Lectures on Nonlinear Dynamics (Understanding Complex Systems) 3031451007, 9783031451003

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Understanding Complex Systems

José Roberto Castilho Piqueira Carlos Eduardo Nigro Mazzilli Celso Pupo Pesce Guilherme Rosa Franzini   Editors

Lectures on Nonlinear Dynamics

Springer Complexity Springer Complexity is an interdisciplinary program publishing the best research and academic-level teaching on both fundamental and applied aspects of complex systems—cutting across all traditional disciplines of the natural and life sciences, engineering, economics, medicine, neuroscience, social and computer science. Complex Systems are systems that comprise many interacting parts with the ability to generate a new quality of macroscopic collective behavior the manifestations of which are the spontaneous formation of distinctive temporal, spatial or functional structures. Models of such systems can be successfully mapped onto quite diverse “real-life” situations like the climate, the coherent emission of light from lasers, chemical reaction-diffusion systems, biological cellular networks, the dynamics of stock markets and of the internet, earthquake statistics and prediction, freeway traffic, the human brain, or the formation of opinions in social systems, to name just some of the popular applications. Although their scope and methodologies overlap somewhat, one can distinguish the following main concepts and tools: self-organization, nonlinear dynamics, synergetics, turbulence, dynamical systems, catastrophes, instabilities, stochastic processes, chaos, graphs and networks, cellular automata, adaptive systems, genetic algorithms and computational intelligence. The three major book publication platforms of the Springer Complexity program are the monograph series “Understanding Complex Systems” focusing on the various applications of complexity, the “Springer Series in Synergetics”, which is devoted to the quantitative theoretical and methodological foundations, and the “Springer Briefs in Complexity” which are concise and topical working reports, case studies, surveys, essays and lecture notes of relevance to the field. In addition to the books in these two core series, the program also incorporates individual titles ranging from textbooks to major reference works. Indexed by SCOPUS, INSPEC, zbMATH, SCImago.

Series Editors Henry D. I. Abarbanel, Institute for Nonlinear Science, University of California, San Diego, La Jolla, CA, USA Dan Braha, New England Complex Systems Institute, University of Massachusetts, Dartmouth, USA Péter Érdi, Center for Complex Systems Studies, Kalamazoo College, Kalamazoo, USA Hungarian Academy of Sciences, Budapest, Hungary Karl J. Friston, Institute of Cognitive Neuroscience, University College London, London, UK Sten Grillner, Department of Neuroscience, Karolinska Institutet, Stockholm, Sweden Hermann Haken, Center of Synergetics, University of Stuttgart, Stuttgart, Germany Viktor Jirsa, Centre National de la Recherche Scientifique (CNRS), Université de la Méditerranée, Marseille, France Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland Kunihiko Kaneko, Research Center for Complex Systems Biology, The University of Tokyo, Tokyo, Japan Markus Kirkilionis, Mathematics Institute and Centre for Complex Systems, University of Warwick, Coventry, UK Ronaldo Menezes, Department of Computer Science, University of Exeter, UK Jürgen Kurths, Nonlinear Dynamics Group, University of Potsdam, Potsdam, Germany Andrzej Nowak, Department of Psychology, Warsaw University, Warszawa, Poland Hassan Qudrat-Ullah, School of Administrative Studies, York University, Toronto, Canada Linda Reichl, Center for Complex Quantum Systems, University of Texas, Austin, USA Peter Schuster, Theoretical Chemistry and Structural Biology, University of Vienna, Vienna, Austria Frank Schweitzer, System Design, ETH Zürich, Zürich, Switzerland Didier Sornette, Institute of Risk Analysis, Prediction and Management, Southern University of Science and Technology, Shenzhen, China Stefan Thurner, Section for Science of Complex Systems, Medical University of Vienna, Vienna, Austria

Understanding Complex Systems Founding Editor: Scott Kelso

José Roberto Castilho Piqueira · Carlos Eduardo Nigro Mazzilli · Celso Pupo Pesce · Guilherme Rosa Franzini Editors

Lectures on Nonlinear Dynamics

Editors José Roberto Castilho Piqueira Escola Politécnica da Universidade de São Paulo São Paulo, Brazil

Carlos Eduardo Nigro Mazzilli Escola Politécnica da Universidade de São Paulo São Paulo, Brazil

Celso Pupo Pesce Escola Politécnica da Universidade de São Paulo São Paulo, Brazil

Guilherme Rosa Franzini Escola Politécnica da Universidade de São Paulo São Paulo, Brazil

ISSN 1860-0832 ISSN 1860-0840 (electronic) Understanding Complex Systems ISBN 978-3-031-45100-3 ISBN 978-3-031-45101-0 (eBook) https://doi.org/10.1007/978-3-031-45101-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

This book is dedicated to: my parents Miguel and Elisa (JRC Piqueira); my daughters Maria Eugênia (in memoriam), Roberta e Paola (CEN Mazzilli); To Sandra, our sons, Fernando (in memoriam), Ricardo and Pedro, and our grand-sons, Francisco, João Vitor, Miguel and Gabriel (CP Pesce); to my family Reinaldo, Maria, Fabiana and Mariah (GR Franzini). The Editors

Foreword

This volume collects a series of essays based on the lectures presented at the widely successful São Paulo School of Advanced Sciences on Nonlinear Dynamics organized by J. R. C. Piqueira, G. R. Franzini, C. P. Pesce and C. E. N. Mazzilli at the Escola Politécnica of the University of São Paulo, Brazil, in the Brazilian winter 2019. The School was conceived as a cross-disciplinary initiative, built around the high quality expertise on nonlinear dynamics available at the Escola Politécnica in different scientific areas (Telecommunications and Control, Offshore Mechanics, Structural and Geotechnical Engineering), and aimed at exposing participants to state-of-art knowledge in the field. Around a valuable local scientific core, the organizers succeeded in securing participation in the School of a number of highly qualified lecturers. The original perspective of the School is now worthily reflected in the content of this volume, which includes a number of important topics in different areas of nonlinear dynamics, addressed by suitably balancing some theoretical aspects of nonlinear dynamics with advanced applications in engineering mechanics and electrical engineering. Eleven chapters are authored by former lecturers (and some coauthors) who can be categorized along two basic lines: (i) some of the best Brazilian scholars of nonlinear dynamics from different academic institutions in São Paulo (University of São Paulo USP), Rio de Janeiro (Instituto de Matemática Pura e Aplicada, Pontifical Catholic University PUC, COPPE-Universidade Federal do Rio de Janeiro UFRJ) and Belo Horizonte (Universidade Federal de Minas Gerais UFMG); (ii) top level scholars from some European universities (University of NoviSad, Serbia, and Obuda University, Budapest, Hungary; Polytechnic University of Marche, Italy; TU Wien, Austria). Covering both research fundamentals and engineering achievements, the overall group of authors highlights the updateness of Brazilian research in nonlinear dynamics, as well as the significance of the longlasting scientific links established by the relevant community with qualified international institutions. The readers of this Foreword will hopefully forgive the writer for quoting himself, which can indeed be considered as an impolite mark of personal scientific opinionatedness. Yet, this is solely aimed at highlighting how nearly all the scientific topics addressed in this volume actually fit into the (of course larger) list of themes identified in a recent review paper as characterizing current advanced vii

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Foreword

research in nonlinear dynamics and its expected developments (ASME J. Comput. Nonlinear Dyn., 17, 080802, 2022). Thus, instead of dwelling in detail on the content of the specific chapters—a job which is possibly left to the book Editors owed to their more proper role and a certainly greater acquaintance—, an attempt is made here to somehow group them into a smaller number of more general themes of interest to the community of scholars of nonlinear dynamics. Indeed, despite the cross-relations certainly existing between topics placed within different groups, in the writer’s opinion they can be broadly referred to as four main general themes: (i) parametric resonance, (ii) nonlinear modal analysis, model reduction and system identification, (iii) synchronization, (iv) strongly nonlinear dynamics. Parametric resonance is a classical theme in the nonlinear vibrations of systems and structures, which has been the subject of a great amount of studies on both the theoretical and the application side. Here, some basic aspects of the underlying mathematical theory are summarized by Alexei Mailybaev, referring to the classical pendulum with a vertically moving support. Floquet stability theory for periodic solutions of linearized systems and its implications for nonlinear stability are dealt with, considering support vertical motions giving rise to two typical ordinary equations (Meissner and Mathieu) and also mentioning other applications in physics. Guilherme Franzini and Guilherme Vernizzi also use the vertically excited pendulum for a preliminary overview of the theme, but they focus on the dynamics of parametrically excited systems and slender structures. Multiple scale methods (and harmonic balance, too) are used to determine transition curves associated with principal parametric instability of the undamped Mathieu equation and the post-critical amplitude of a class of ensuing nonlinear systems (i.e., the Duffing-Mathieu-Morrison oscillator describing the parametric excitation of a vertical slender structure immersed in a fluid). Referring to marine structures as a plentiful field for the occurrence of parametric excitation, an overview of recent contributions is presented, and outcomes of reduced order modeling are discussed in comparison with finite element simulations. Nonlinear modal analysis, nonlinear model reduction and nonlinear system identification are certainly distinct conceptual issues in nonlinear dynamics, which are however somewhat correlated with each other, though to a different extent. In the last few decades, all of them have been the subject of meaningful advancements, and further ones are currently being obtained mostly in connection with the transition from archetypal or anyway minimal reduced models of systems and structures to actual multi-degree-of-freedom (mdof) models, this being a major challenging topic for the future of nonlinear dynamics. Within this wide framework, Alois Steindl dwells on normal forms and nonlinear normal modes (NNMs) in substantially theoretical terms. Normal form theory helps to simplify linear and nonlinear dynamical systems by reducing the number of terms in the equations, and by introducing symmetries. The main ideas for NNMs are also introduced, focusing on approximate calculation and the relation with symmetry properties which also entail localization phenomena. A rotational symmetric system is investigated through the equivariant bifurcation theory, yielding the list of expected solution types in a systematic way. Nonlinear normal modes for the analysis of mdof systems and the derivation of proper reduced-order models (ROMs) for forced dynamics making use of only a few

Foreword

ix

nonlinear modes are dealt with by Paulo Goncalves. Based on their modern interpretation as two-dimensional invariant manifolds in phase space, NNMs are approximated by asymptotic expansions in terms of a pair of master coordinates or through a Galerkin expansion. Issues of identification and stability, and numerical continuation techniques to obtain frequency-energy plots and frequency-amplitude relations are discussed, with reference to illustrative examples of two-dof systems. System identification is the ‘art’ of building mathematical models of a dynamical system from measured data. Luis Aguirre deals with the matter based on a kind of ‘philosophical framework’ which, although being admittedly subjective, aims and succeeds at providing the newcomer with an introduction to some real challenges of this fascinating field. A typical system identification problem is divided into five steps, i.e., (i) testing and data collection, (ii) choice of model class, (iii) structure selection, (iv) parameter estimation, and (v) model validation, with preliminary discussions made and a clear bias towards nonlinear system identification. Black-box and grey-box techniques, respectively building mathematical models based on the solely available set of measured data or exploiting other sources of information about the dynamical systems, are considered. Carlos Mazzilli, Eduardo Ribeiro and Breno Mendes provide a comprehensive discussion about the modal asynchronicity (or localisation) in structural systems, departing from a Ziegler’s column with twodof which highlights how even simple systems can display asynchronous modes. Admissibility conditions for modal asynchronicity are discussed in a series of linear systems, and also in the nonlinear realm, showing the occurrence of both nearly asynchronous and exactlyasynchronous NNMs. The technological potentiality of modal asynchronicity for vibration control and energy harvesting is also addressed. A completely distinct perspective on synchronization (i.e., temporal instead of spatial, as of the previous one) is offered by José Piqueira, who dwells on phase and frequency problems in telecommunication devices or networks, where a coherent modulation has to be secured. In a great number of practical applications, isolated or networked phaselocked loops are responsible for correct recovering of the time basis, and synchronizing the processes. Among different clock distribution strategies, two topologies for time signal distribution are compared—i.e., one-way, not considering clock feedback, and two-way master-slave, with the slave nodes providing clock feedback to the master—, along with the related performance. Topics addressed in four more chapters can be roughly categorized under another challenging perspective of the development of nonlinear dynamics, linked with the need to go more in-depth into the description and understanding of strongly nonlinear phenomena. Yet, within such a tentatively unifying framework—which, indeed, also includes the previous problem of phase and frequency synchronization—, distinct perspectives can be identified. An analytical perspective, aimed at searching exact or approximate solutions for strongly nonlinear vibrations, underlies the contribution by Livija Cveticanin, who generalizes to the Van der Pol oscillator with a truly (or essentially) and strongly nonlinear term, the Krylov-Bogoliubov method originally introduced for solving weakly nonlinear equations. Cases of a time variable parameter or a slowly variable mass are dealt with, along with the steady state motion of a forced strongly nonlinear oscillator, for which Melnikov criterion and delayed feedback control are used to determine conditions

x

Foreword

for the existence of chaos and its suppression, respectively. Analytical mechanics is the general framework referred to by Celso Pesce, Renato Orsino and Leandro Silva for readdressing the somewhat specific topic of nonlinear dynamics of variable mass systems, with consideration of both systems of particles and nonmaterial volumes, and for exploring the developed general formulation in archetypical nonlinear oscillators with variable, and especially position-dependent, mass. Then, two statistical approaches are used for treating problems in nonlinear fluid dynamics, with emphasis on the water column dynamics in the free surface piercing pipe excited by random gravitational waves. Marcelo Savi presents a combined theoretical/computational/ phenomenological overview of chaotic dynamics, starting from the dynamical system background and presenting mathematical representation and the concept of stability. Chaos is characterized in terms of the contraction-expansion fold process associated with horseshoe transformation, and some of its main aspects are discussed, defining chaotic and fractal attractors, summarizing routes to chaos ensuing from local and global bifurcations, and referring to Lyapunov exponents as a proper diagnostic tool. Finally, a clear safety/design perspective can be recognized in Stefano Lenci’s overview of dynamical integrity, which is the branch of nonlinear dynamics aimed at checking the robustness, or practical stability, of solutions of evolutionary systems governed by differential equations or maps. Despite being developed in the engineering realm, ideas and results apply to any dynamical system, irrespective of the problem of interest. The classical concept of stability is recalled to properly frame searched information about the system’s global dynamics, which is of great relevance for application purposes. The main elements of a robustness analysis are overviewed, and the regularity of the integrity profiles is specifically dealt with. Of course, the topics dealt with in this book do not exhaust the large and highly varied basket of themes of theoretical and application importance in such a wide and transdisciplinary area as nonlinear dynamics. Nonetheless, they fully succeed in providing the readers of the book, which is certainly of interest not only for young scientists but also for more acquainted scholars, with an effective and updated overview of the advanced research which is being conducted in the area at the international level, as well as of a few challenging themes for generations of young scientists who are now entering this fascinating field of studies. Indeed, its great potential in terms of enhanced reliability, performance and effectiveness of systems and structures, as well as possible formulation of novel paradigms for safe and aware design, is certainly far from being fully grasped and/or adequately exploited. Rome, Italy June 2022

Giuseppe Rega Professor Emeritus

Preface

This book assembles the contents of lectures presented in the São Paulo School of Advanced Sciences on Nonlinear Dynamics, held at Escola Politécnica, University of São Paulo, from July 29 to August 9, 2019, with the support of FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo). The School gathered 121 students, from undergraduates to post-docs, 53 of them foreigners, and it was rated by them as a complete success in reaching its objectives, namely: to supply the students basic knowledge and perspectives for research in nonlinear dynamics and to display the potentialities to carry out research in nonlinear dynamics within the State of São Paulo, in particular at Escola Politécnica, with a view to its infrastructure and available expertise. The organizers committed themselves to editing this book to enhance the benefits of accessing this valuable material to a larger public of prospective researchers in the field of nonlinear dynamics. Along its 11 chapters, fundamental and novelty topics are gathered, ranging from formulation to methods of analysis, from phenomena to case studies and applications. The 11 chapters of the book are organized into four large groups: (i) parametric resonance, (ii) nonlinear modal analysis and model reduction, (iii) synchronization and (iv) strongly nonlinear dynamics, as masterfully classified and described in the Foreword, kindly written by Prof. Giuseppe Rega, an undisputed authority on the field of nonlinear dynamics and its applications to engineering. These groups are naturally intertwined, mixing approaches and themes, from fundamental content to applied ones; from introductory to more advanced subjects. The first group is represented by Chaps. 1 and 2, as introductions to the theory of parametric resonance and the dynamics of parametrically excited slender structures. The second group is covered by Chaps. 3–5. Chapter 3 brings a thorough account on normal forms and Chap. 4 goes into the realm of nonlinear normal modes and reduced order modeling techniques. Chapter 5 provides a clear introduction to nonlinear system identification.

xi

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Preface

The third group is the subject of the Chaps. 6 and 7. Chapter 6 covers asynchronous modes of structural vibration and Chap. 7 treats master-slave topologies for time signal distribution within a distinct perspective of synchronous systems. The last four chapters treat the fourth group. Chapter 8 considers the nonlinear dynamics of variable mass oscillators, from the point of view of analytical mechanics and statistical methods. Chapter 9 brings a detailed account on the application of advanced analytical methods for solving strong nonlinear vibration problems. Chapter 10 provides an overview of chaos theory and, finally, Chap. 11 gifts the reader, by ending this book with a solid foundation in dynamic integrity from the perspectives of safety and design. The organizers are grateful to the authors who contributed to the course and to the book; to Vitor Schwenck Franco Maciel and Guilherme Vernizzi for the preparation of the camera-ready text; and to the Departments of Structural and Geotechnical Engineering (PEF), Mechanical Engineering (PME) and Telecommunication and Control Engineering (PTC) at Escola Politécnica, for the academic and financial support that made it possible to publish this book. Our special recognition to FAPESP, for enabling the two weeks of unique experience lived during the São Paulo School of Advanced Sciences in Nonlinear Dynamics, vividly and friendly shared by lecturers and students. São Paulo January 2023

José Roberto Castilho Piqueira Carlos Eduardo Nigro Mazzilli Celso Pupo Pesce Guilherme Rosa Franzini

Acknowledgements

The authors thank the financial support provided by The São Paulo Research Foundation (FAPESP) through grant 2018/16829-8, which allowed the realization of the São Paulo School of Advanced Sciences on Nonlinear Dynamics from July 29 to August 9, 2019, gathering 121 students and 12 lecturers from worldwide. Important organizational support was also provided by Escola Politécnica of the University of São Paulo, in particular by the following departments: Telecommunication and Control Engineering (PTC), Structural and Geotechnical Engineering (PEF), and Mechanical Engineering (PME). All the students and lecturers who took part in this project are also acknowledged, with special thanks to those involved in the organization of the event. The editors also acknowledge the support of the National Council for Scientific and Technological Development (CNPq), through grants 302883/20185 (JRC Piqueira), 301050/2018-0 (CEN Mazzilli) 307995/2022-4 (CP Pesce) and 305945/2020-3 (GR Franzini). Last, but not least, the editors would like to thank Guilherme Vernizzi and Vitor Maciel for their invaluable support in the preparation and organization of this book.

xiii

Contents

1

Brief Introduction to the Theory of Parametric Resonance . . . . . . . . Alexei A. Mailybaev

2

An Introduction to Parametrically Excited Systems and Their Importance on the Dynamics of Slender Structures . . . . . . . . . . . . . . . Guilherme Rosa Franzini and Guilherme Jorge Vernizzi

1

27

3

Normal Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alois Steindl

57

4

Nonlinear Normal Modes and Reduced Order Models . . . . . . . . . . . . 105 Paulo Batista Gonçalves

5

An Introduction to Nonlinear System Identification . . . . . . . . . . . . . . 133 Luis A. Aguirre

6

Asynchronous Modes of Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Carlos E. N. Mazzilli, Eduardo A. R. Ribeiro, and Breno A. P. Mendes

7

Comparing Master-Slave Topologies for Time Signal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 José Roberto Castilho Piqueira

8

Nonlinear Dynamics of Variable Mass Oscillators . . . . . . . . . . . . . . . . 217 Celso P. Pesce, Renato M. M. Orsino, and Leandro S. P. Silva

9

Generalized Krylov-Bogoliubov Method for Solving Strong Nonlinear Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Livija Cveticanin

10 Chaos Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Marcelo A. Savi 11 Dynamical Integrity and Its Background . . . . . . . . . . . . . . . . . . . . . . . . 301 Stefano Lenci xv

Contributors

Luis A. Aguirre Departamento de Engenharia Eletrônica, Universidade Federal de Minas Gerais, Minas Gerais, Brazil Livija Cveticanin Obuda University, Becsi ut 96/B, Hungary Guilherme Rosa Franzini Offshore Mechanics Laboratory, Escola Politécnica da USP (LMO/EPUSP), São Paulo, Brazil Paulo Batista Gonçalves Pontifical Catholic University of Rio de Janeiro, PUCRio, Rio de Janeiro, Brazil Stefano Lenci Polytechnic University of Marche, Ancona, Italy Alexei A. Mailybaev Instituto de Matemática Pura e Aplicada, IMPA, Rio de Janeiro, Brazil Carlos E. N. Mazzilli University of São Paulo, São Paulo, Brazil Breno A. P. Mendes University of São Paulo, São Paulo, Brazil Renato M. M. Orsino University of São Paulo, São Paulo, Brazil Celso P. Pesce University of São Paulo, São Paulo, Brazil José Roberto Castilho Piqueira Escola Politécnica da Universidade de São Paulo, Avenida Prof Luciano Gualberto, trav. 3 - 158, São Paulo, Brazil Eduardo A. R. Ribeiro University of São Paulo, São Paulo, Brazil Marcelo A. Savi Center for Nonlinear Mechanics—Department of Mechanical Engineering—COPPE—Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil

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Contributors

Leandro S. P. Silva University of Adelaide, Adelaide, Australia Alois Steindl Institute for Mechanics and Mechatronics, TU Wien, Vienna, Austria Guilherme Jorge Vernizzi Offshore Mechanics Laboratory, Escola Politécnica da USP (LMO/EPUSP), São Paulo, Brazil

Chapter 1

Brief Introduction to the Theory of Parametric Resonance Alexei A. Mailybaev

Abstract In these lecture notes, we present a basic mathematical theory of parametric resonance. This includes the Floquet stability theory for periodic systems of linear differential equations and its implications for nonlinear stability. As applications to mechanical systems, we consider a pendulum with a moving support. In the end we discuss other applications in physics. We refer the reader for further studies to the books [1, 2, 6].

1.1 Pendulum with a Moving Support Consider a pendulum with a support moving vertically; see Fig. 1.1. The vertical position of the pendulum is given by a periodic function of time .

h(t) = h(t + T ),

Ω=

2π , T

(1.1)

where .T is a period and .Ω is the corresponding frequency. The function .h(t) is assumed to be smooth or piecewise smooth with a finite number of jumps. Position of the the system is defined by the angle .ϕ between the pendulum and the vertical axis. The vertical position .ϕ = 0 is an equilibrium. Let us define the coordinate system (fixed in space) with the horizontal axis .x and vertical axis . y. Coordinates of the moving mass for a pendulum of length .l are given by . x = l sin ϕ, y = h(t) + l(1 − cos ϕ). (1.2) The corresponding temporal derivatives denoted by the dots are .

x˙ = lϕ˙ cos ϕ,

˙ + lϕ˙ sin ϕ. y˙ = h(t)

(1.3)

A. A. Mailybaev (B) Instituto de Matemática Pura e Aplicada, IMPA, Rio de Janeiro, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. R. Castilho Piqueira et al. (eds.), Lectures on Nonlinear Dynamics, Understanding Complex Systems, https://doi.org/10.1007/978-3-031-45101-0_1

1

2

A. A. Mailybaev

Fig. 1.1 Pendulum with a support moving vertically with a frequency .Ω

The Lagrangian function of this system is written as the difference .

L(ϕ, ϕ) ˙ = T − U,

(1.4)

of the kinetic and potential energies .

T =

) m( 2 x˙ + y˙ 2 , 2

U = mg y,

(1.5)

where .g is the acceleration of gravity. Equations of motion are given by the Euler– Lagrange equation ∂L d ∂L − = 0, . (1.6) dt ∂ ϕ˙ ∂ϕ where .d/dt is the material derivative taken along the system trajectory. Expressions (1.1)–(1.5) yield the final expression for the Lagrangian as ( L = ml2

.

) ϕ˙ 2 h˙ ϕ˙ g + sin ϕ + cos ϕ , 2 l l

(1.7)

where we omitted all terms that do not depend on .ϕ or .ϕ˙ and, therefore, will not appear in the final Eq. (1.6). Using this function in the Euler–Lagrange equation (1.6) yields ( ) (˙ ) h˙ h ϕ˙ g d ϕ˙ + sin ϕ − cos ϕ − sin ϕ = 0, . (1.8) dt l l l where we dropped the common pre-factor .ml2 . Taking the material derivative, we derive the final equation of motion in the form

1 Brief Introduction to the Theory of Parametric Resonance

3

( ) g h¨ .ϕ ¨+ 1+ sin ϕ = 0. l g

(1.9)

¨ This equation has a periodic coefficient .h(t), which will lead to the phenomenon of parametric resonance. It is convenient to introduce an extra dependent variable .ψ = ϕ˙ and reduce Eq. (1.9) to the system of first-order differential equations y˙ = g(y, t),

(1.10)

.

where the vector .y and the function .g(y, t) are defined as ⎛

( ) ϕ .y = , ψ

ψ

⎞

( ) ⎠. h¨ g(y, t) = ⎝ g 1+ sin ϕ − l g

(1.11)

This system has an equilibrium .ϕ = ψ = 0 corresponding to the vertical position of the pendulum.

1.2 Poincaré Map Motivated by the example in the previous section, we consider a general system of the form ˙ = g(y, t), .y (1.12) where .y ∈ Rn and the right-hand side is a periodic function of time, g(y, t) = g(y, t + T ),

.

(1.13)

with period .T . Additionally, we assume that the system has an equilibrium (fixedpoint) solution .y(t) ≡ y∗ , (1.14) which implies that g(y∗ , t) = 0,

.

(1.15)

for all .t ∈ R. This system appears in various applications, where periodic timedependence of parameters is caused by external oscillations. For example, one may think of stability of structures under the action of waves, e.g., earthquakes or sea storms. For the analysis of solutions .y(t), it is convenient to introduce a concept of Poincaré map. This is a function

4

A. A. Mailybaev

Fig. 1.2 Poincaré section is defined by considering the solution at integer multiples of the period .T

f : Rn |→ Rn ,

(1.16)

y(T ) = f(y(0)),

(1.17)

.

which is defined by the relation .

i.e., given the initial condition .y(0), it returns the solution .y(T ) after one period; see Fig. 1.2. Let us denote .yk = y(tk ), tk = kT, k ∈ Z, (1.18) which correspond to the solution evaluated at integer multipliers of the period. From the periodicity condition (1.13), it follows that y

. k+1

= f(yk ),

(1.19)

for all .k. Relation (1.19) defines the system with the “discrete” time .k ∈ Z, and the solution .yk can be seen as a sequence of “snapshots” of the original continuoustime evolution (1.12) taken times .tk ; see Fig. 1.2 for an illustration. Condition (1.14) implies that .y∗ is a fixed point of the Poincaré map: y = f(y∗ ).

. ∗

(1.20)

1.3 Linearized System and the Floquet Matrix Let us consider oscillations near the equilibrium (1.14). Defining the vector .x, which describes small deviations from .y∗ , we write y = y∗ + x.

.

(1.21)

1 Brief Introduction to the Theory of Parametric Resonance

5

Then, the function .g(y, t) can be represented using the Taylor expansion in the form .g(y, t) = g(y∗ + x, t) = g(y∗ , t) + G(t)x + o(||x||), (1.22) where ⎛ ∂g1

∂g1 ∂ y1 ∂ y2 ∂g2 ∂g2 ∂ y1 ∂ y2

⎜ ⎜ .G(t) = ⎜ . ⎜. ⎝.

.. .

∂gn ∂gn ∂ y1 ∂ y2

· · · ∂∂gyn1 · · · ∂∂gyn2 . . .. .. n · · · ∂g ∂ yn

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

,

(1.23)

y=y∗

is the Jacobian matrix of .g(y, t) evaluated at the equilibrium point. In the last expression of (1.22), the first term vanishes by the condition (1.15), and the last correction term .o(||x||) is small compared to .G(t)x. Neglecting this correction term in (1.22), Eq. (1.12) yields the linearized system x˙ = G(t)x.

.

(1.24)

By the periodicity condition (1.13), the matrix .G(t) depends on time periodically with the same period .G(t) = G(t + T ). (1.25) As an example, let us consider the system (1.11) with the equilibrium .y∗ = 0. The matrix of the linearized system is easily derived as ⎛

⎞ 0 1 ( ) ⎠. g h¨ .G(t) = ⎝ − 1+ 0 l g

(1.26)

One can linearize the discrete-time system (1.19) similarly to how we linearized the continuous-time system. For this purpose, we define the small deviation vectors .xk from the fixed point as .yk = y∗ + xk . (1.27) Using the Taylor expansion, the Poincaré map is written as f(yk ) = f(y∗ + xk ) = f(y∗ ) + Fxk + o(||xk ||) = y∗ + Fxk + o(||xk ||),

.

where

(1.28)

6

A. A. Mailybaev

⎛ ∂ f1

∂ f1 ∂ y1 ∂ y2 ∂ f2 ∂ f2 ∂ y1 ∂ y2

⎜ ⎜ .F = ⎜ . ⎜. ⎝.

.. .

∂ fn ∂ fn ∂ y1 ∂ y2

· · · ∂∂ yf1n · · · ∂∂ yf2n . . .. .. · · · ∂∂ yfnn

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

,

(1.29)

y=y∗

is the Jacobian matrix of .f(y) evaluated at the fixed point. The matrix .F does not depend on time and is called the Floquet matrix. Substituting .yk+1 = y∗ + xk+1 and (1.28) into (1.19) and neglecting the nonlinear higher-order term .o(||xk ||), we obtain the linearized discrete-time system in the form x

. k+1

= Fxk .

(1.30)

The relation between the continuous and discrete time variables (1.18) induces analogous relation for the linearized variables x = x(tk ), tk = kT, k ∈ Z.

. k

(1.31)

This relation suggests a practical method for computing the Floquet matrix .F as follows. Let us consider the Cauchy problem .

˙ = G(t)X, X

X(0) = I,

(1.32)

for the .n × n matrix function .X(t), which is called the fundamental solution. Here, every column of.X(t) is a solution of the linearized system (1.24) for initial conditions equal to the corresponding column of the identity matrix ⎛

1 ⎜0 ⎜ .I = ⎜ . ⎝ ..

0 1 .. .

··· ··· .. .

⎞ 0 0⎟ ⎟ .. ⎟ . .⎠

(1.33)

0 0 ··· 1

One can see that x(t) = X(t)x0 ,

.

(1.34)

is a solution of the linearized system (1.24) for initial condition .x(0) = x0 . This yields .x1 = x(T ) = X(T )x0 . (1.35) Comparing this relation with (1.30) for .k = 0, we obtain the Floquet matrix as F = X(T ).

.

(1.36)

1 Brief Introduction to the Theory of Parametric Resonance

7

This means that the Floquet matrix can be found by integrating the linearized system in one period.

1.4 Linear Discrete-Time Dynamical System In this section, we describe a general solution of the system x

. k+1

= Fxk .

(1.37)

For this purpose, let us consider the eigenvalue problem Fu = ρu,

.

(1.38)

where .ρ ∈ C is an eigenvalue and .u ∈ Cn is a corresponding eigenvector. Notice that both .ρ and .u may take complex values. This is no problem, because system (1.37) is linear and, hence, real and imaginary parts of complex solutions yield real solutions to the same problem. Each pair .ρ and .u defines an explicit solution x = ρ k u.

. k

(1.39)

Because of this form of solution, eigenvalues of the Floquet matrix are also called the multipliers. When the multiplier .ρ is real, the eigenvector .u can also be taken real, which yields a real solution (1.39). When ρ = |ρ|eiϕ = |ρ|(cos ϕ + i sin ϕ),

.

(1.40)

is a complex number, we can construct real solutions by taking real and imaginary parts of (1.39). This yields two different solutions x = |ρ|k [Re u cos(kϕ) − Im u sin(kϕ)] ,

(1.41)

x = |ρ|k [Re u sin(kϕ) + Im u cos(kϕ)] .

(1.42)

. k

and . k

Exactly the same pair of solution is obtained for the complex conjugate eigenvalue ρ and eigenvector .u. Thus the two solutions (1.41) and (1.42) are associated with the complex conjugate pair .ρ and .ρ. Writing Eq. (1.38) as .(F − ρI)u = 0, (1.43)

.

we conclude that the non-trivial solution .u exists if and only if .

det(F − ρI) = 0.

(1.44)

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A. A. Mailybaev

This is the so-called characteristic equation. The left-hand side of this equation is a polynomial of degree .n, which has .n complex roots .ρ1 , . . . , ρn . In the case, when all the roots are simple (distinct), there are .n corresponding eigenvector .u1 , . . . , un , which are linearly independent. Taking the linear combination of solutions (1.39) for all multipliers, we obtain a general solution x =

n ∑

. k

c j ρ kj u j ,

(1.45)

j=1

where .c j are arbitrary coefficients. These coefficients are determined uniquely by the initial condition n ∑ .x0 = cjuj, (1.46) j=1

because the vectors .u1 , . . . , un form a basis. This solution can be written in the real form by using expressions (1.41) and (1.42) instead of (1.39) for each complex conjugate pair of multipliers. Example 1.1 Let is consider the matrix ( ) 21 .F = . 12

(1.47)

Characteristic Eq. (1.44) for this matrix takes the form ρ 2 − 4ρ + 3 = 0,

(1.48)

ρ = 1, ρ2 = 3.

(1.49)

.

and yields two multipliers . 1

The corresponding eigenvectors are found by solving the system (1.43) for each multiplier as ( ) ( ) 1 1 .u1 = (1.50) , u2 = . −1 1 The general solution (1.45) is written in the form ( x = c1

. k

( ) ) 1 1 + c2 3k , −1 1

(1.51)

with arbitrary real coefficients .c1 and .c2 . Example 1.2 Let is consider the matrix (

) 0 1 .F = . −1 0

(1.52)

1 Brief Introduction to the Theory of Parametric Resonance

9

Characteristic Eq. (1.44) for this matrix takes the form ρ 2 + 1 = 0,

.

(1.53)

and yields two complex conjugate multipliers ρ = i = eiπ/2 , ρ2 = −i = e−iπ/2 .

. 1

(1.54)

The corresponding eigenvectors are found by solving the system (1.43) as u =

. 1

( ) ( ) 1 1 , u2 = . i −i

(1.55)

The general solution (1.45) is written in the complex form ( ) ( ) 1 1 + c2 (−i)k , i −i

x = c1 i k

. k

(1.56)

with the complex coefficients.c1 and.c2 . In order to obtain a real solution, one can take c = (a − ib)/2 and .c2 = (a + ib)/2 with real numbers .a and .b. Then, expression (1.56) takes the form

. 1

( ) ] [( ) ( ) ] [( ) kπ kπ kπ kπ 0 1 0 1 − sin +b sin + cos . (1.57) .xk = a cos 1 0 1 0 2 2 2 2 The two solutions multiplied by .a and .b in this expression are exactly the two real solutions (1.41) and (1.42). Example 1.3 Let is consider the matrix ( F=

.

) 3 1 . −1 1

(1.58)

Characteristic Eq. (1.44) for this matrix takes the form ρ 2 − 4ρ + 4 = 0,

(1.59)

ρ = ρ2 = 2.

(1.60)

.

and yields two coincident roots . 1

From the system (1.43) one obtains a single eigenvector (

) 1 .u = . −1

(1.61)

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A. A. Mailybaev

In this case the general solution cannot be constructed in the form (1.45), because the eigenvectors do not form a basis. This example shows that the general solution gets more complicated, when some of the eigenvalues are multiple.

1.5 Multiple Eigenvalues and Jordan Chains Let us consider now the general case, when the eigenvalue .ρ is not simple, i.e., it appears .m a > 1 times as the root of the characteristic Eq. (1.44). The number .m a is called the algebraic multiplicity (or simply the multiplicity) of the eigenvalue. Recall that the eigenvalue is called simple if .m a = 1. The number .m g of linear independent eigenvectors corresponding to .ρ is called a geometric multiplicity. From the linear algebra, we know that 1 ≤ mg ≤ ma .

.

(1.62)

In the case .m g = m a > 1, the eigenvalue .ρ is called semi-simple: it is multiple, but has the same number of linearly independent eigenvectors as the number of roots of characteristic equation. If all roots are simple or semi-simple, one can still write the general solution in the form (1.45), because it is possible to form a basis with eigenvectors. The solution becomes more complicated if .m g < m a . In this case extra solutions must be determined. We say that linearly independent vectors .u1 , . . . , ul form a Jordan chain corresponding to .ρ if they satisfy the system of equations .

Fu1 = ρu1 ,

(1.63)

Fu2 = ρu2 + u1 , .. .

(1.64)

Ful = ρul + ul−1 ,

(1.65)

Ful+1 /= ρul+1 + ul ,

(1.66)

while .

for any vector .ul+1 . Here .u1 is the eigenvector and the other vectors .u2 , . . . , ul are called associated vectors (or generalized eigenvectors). Note that these vectors are real for the real .ρ and complex for the complex .ρ. Let us define a .n × l matrix .U with the Jordan chains vectors takes as columns: U = [u1 u2 · · · ul ].

.

(1.67)

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11

Then the chain (1.63)–(1.65) can be written as a single relation FU = UJ,

(1.68)

.

⎛

where

ρ 1

⎜ ⎜ ρ ... .J = ⎜ ⎜ .. ⎝ .

⎞ ⎟ ⎟ ⎟, ⎟ 1⎠ ρ

(1.69)

is the .m × m matrix called the Jordan block. It has the eigenvalue .ρ on the main diagonal, unity on the next (upper) diagonal and zeros otherwise. Proposition 1.1 If .u1 , . . . , ul are the Jordan chain vectors, then the following .l expressions x = ρ k u1 , xk = ρ k u2 + kρ k−1 u1 , .. .

(1.70) (1.71)

. k

xk = ρ k ul + kρ k−1 ul−1 + · · · +

k! ρ k−l u1 , l!(k − l)!

(1.72)

are solutions of system (1.37). This statement can be verified by the direct substitution. For example, for the second expression, we have Fxk = F(ρ k u2 + kρ k−1 u1 ) = ρ k (ρu2 + u1 ) + kρ k u1 . = ρ k+1 u2 + (k + 1)ρ k u1 = xk+1 ,

(1.73)

where we used the Eqs. (1.63) and (1.64). Note that, for .k = 0, expressions of Proposition 1.1 become . 0

x = u1 ,

(1.74)

x0 = u2 , .. .

(1.75)

x0 = ul ,

(1.76)

which means that these.l solutions are linearly independent. We showed that a Jordan chain of length .l defines .l linearly independent solutions, which all stay in the linear subspace spanned by the vectors of the Jordan chain.

12

A. A. Mailybaev

One of central results of the linear algebra is the Jordan normal form theorem; see e.g. [3]. It says that any matrix can be reduced to the Jordan canonical form by a non-singular complex matrix .C, i.e., ⎛ ⎜ C−1 FC = ⎝

.

J1

⎞ ..

⎟ ⎠,

.

(1.77)

JA where the right-hand side is a block-diagonal matrix and .J1 , . . . , J A are Jordan blocks that contain all eigenvalues of the matrix .F. Splitting columns of the matrix .C = [U1 · · · U A ] accordingly, one can see that each block .Ua with .a = 1, . . . , A defines a Joran chain (1.67) for the Jordan block .Ja ; see Eq. 1.68. This observation leads to the following result: Theorem 1.1 Consider a basis constructed with the vectors of Jordan chains for all eigenvalues, as provided by the Jordan form theorem. Then, the general solution of system (1.37) is obtained by taking a linear combination of solutions described in Proposition 1.1 for all eigenvalues and corresponding Jordan chains. Note that the geometric multiplicity .m g of a given eigenvalue .ρ defines a number of corresponding Jordan chains (Jordan blocks) in the Jordan normal form (1.77). Let .l1 , . . . , lm g be lengths of such Jordan chains (sizes of Jordan blocks). Then, the algebraic multiplicity is equal to the total number of vectors in these chains: .m a = l1 + · · · + lm g . Example 1.4 We have shown in Example 3 that the matrix (1.58) has a multiple eigenvalue .ρ = 2 of algebraic multiplicity .m a = 2 and geometric multiplicity .m g = 1. This means that there must be a Jordan chain, which consists of the eigenvector .u1 and the associated vector .u2 . By solving Eqs. (1.63) and (1.64) for the matrix (1.58), we obtain ( ) ( ) 1 1 .u1 = (1.78) , u2 = . −1 0 The general solution is obtained as a linear combination of solutions (1.70) and (1.71): ( ) ( ) 1 k k−1 k 1 .xk = (c1 2 + c2 k2 ) + c2 2 (1.79) . −1 0

1.6 Stability Theory We now can formulate the concepts of Lyapunov stability for equilibrium solutions in continuous- and discrete-time systems. Definition 1.1 The equilibrium solution .y(t) ≡ y∗ of system (1.12) is called

1 Brief Introduction to the Theory of Parametric Resonance

13

• stable if for any .ε > 0 there exists .δ > 0, such that solutions starting .δ-close to the equilibrium remain .ε-close, i.e., ||y(t) − y∗ || < ε,

.

(1.80)

for all times .t > 0 and all initial conditions satisfying ||y(0) − y∗ || < δ.

.

(1.81)

• asymptotically stable if it is stable with the additional condition that .

lim y(t) = y∗ ,

t→+∞

(1.82)

for all initial conditions satisfying (1.81). The concept of stability is formulated similarly for discrete-time systems. Definition 1.2 The equilibrium solution .yk ≡ y∗ of system (1.19) is called • stable if for any .ε > 0 there exists .δ > 0, such that solutions starting .δ-close to the equilibrium remain .ε-close, i.e., ||yk − y∗ || < ε,

.

(1.83)

for all .k > 0 and all initial conditions satisfying ||y0 − y∗ || < δ.

.

(1.84)

• asymptotically stable if it is stable with the additional condition that .

lim yk = y∗

k→+∞

(1.85)

for all initial conditions satisfying (1.84). In our case, these two definitions are in fact equivalent: Proposition 1.2 The equilibrium solution .y(t) ≡ y∗ of system (1.12) is (asymptotically) stable if and only if the equilibrium solution .yk ≡ y∗ is (asymptotically) stable for the Poincaré map (1.19). This proposition follows from the continuous dependence of solutions of ordinary differential equations on initial conditions. Now, let us formulate the stability criteria. The first theorem provides the complete description of stability in the case of linear systems.

14

A. A. Mailybaev

Fig. 1.3 Location of multipliers for the stability of a linear discrete-time system

Theorem 1.2 (Floquet) The trivial solution .xk ≡ 0 of the linear system (1.37) is • stable if and only if .|ρ| ≤ 1 for all eigenvalues of the Floquet matrix and all eigenvalues with .|ρ| = 1 are simple or semi-simple (there are no nontrivial Jordan chains). • asymptotically stable if and only if .|ρ| < 1 for all eigenvalues of the Floquet matrix. Figure 1.3 provides the graphical illustration to the above theorem. We will not provide the detailed proof here. The idea is to use the explicit general solution from Theorem 1.1. All terms in this solution have the form .k m ρ k for the discrete time .k = 0, 1, 2, . . .. If .|ρ| > 1, then such a term tends to infinity as .k → +∞, because of the dominant exponential part. Similarly, if .|ρ| < 1, then this term vanishes as .k → +∞. When.|ρ| = 1, the term remains bounded only if.m = 0 (no Jordan chains) and grows as a power law otherwise. The second theorem refers to the equilibrium of the nonlinear system. Theorem 1.3 (Lyapunov) Consider the equilibrium solution .yk ≡ y∗ of system (1.19). • If .|ρ| < 1 for all eigenvalues of the Floquet matrix, then the equilibrium is asymptotically stable. • If .|ρ| > 1 for at least one eigenvalue of the Floquet matrix, then the equilibrium is unstable. For the proof of this theorem we refer to graduate courses on ordinary differential equations; see e.g. [1]. There is an important issue in Theorem 1.3. Namely, it does not cover the case, when all eigenvalues satisfy the condition.|ρ| ≤ 1 with at least one

1 Brief Introduction to the Theory of Parametric Resonance

15

of them belonging to the unit circle, .|ρ| = 1. This case is referred to as the special case of Lyapunov, and this is the case when nonlinear higher-order terms become important for stability. In this special case, one cannot blindly rely on the linearized system in the stability analysis.

1.7 The Meissner Equation We now return to the example of Sect. 1.1 and consider a special type of the function h(t) corresponding to vertical motion of the pendulum support. Let us consider the function { Ω 2 Δ, 0 ≤ t < T /2, ¨ .h(t) = (1.86) −Ω 2 Δ, T /2 ≤ t < T,

.

which is extended periodically with the period .T (and frequency .Ω = 2π/T ) to larger times. This function corresponds to piecewise constant acceleration of the support, where .Δ is a parameter proportional to the amplitude of .h(t). Function (1.86) corresponds to the periodic support function .h(t), which consists of parabolic segments and has approximately a sinusoidal form as shown in Fig. 1.4. We introduce the rescaled time τ = Ωt, Ω =

.

2π , T

(1.87)

where .Ω is the frequency of the support oscillations. Then the equation of motion (1.9) for the pendulum is written in the form .

d 2ϕ + [a + bH (τ )] sin ϕ = 0, dτ 2

Fig. 1.4 Acceleration (left panel) and position (right panel) of the pendulum support

(1.88)

16

A. A. Mailybaev

{

where .

H (τ ) =

and a=

.

1, −1,

0 ≤ τ < π; π ≤ τ < 2π,

g Δ , b= . 2 lΩ l

(1.89)

(1.90)

This system has an equilibrium .ϕ(τ ) ≡ 0, and the corresponding linearized equation is obtained by the substitution .sin ϕ ≈ ϕ as .

d 2ϕ + [a + bH (τ )] ϕ = 0. dτ 2

(1.91)

Coefficients of Eqs. (1.88) and (1.91) are periodic functions of .τ with the period 2π .

.

1.7.1 Floquet Matrix First, let us introduce an extra dependent variable .ψ = dϕ/dτ , and reduce (1.91) to the canonical (first-order) form dx = G(τ )x, dτ

(1.92)

( ) ( ) ϕ 0 1 .x = , G(τ ) = . ψ −a − bH (τ ) 0

(1.93)

.

where

The matrix .G(τ ) is periodic with the period .T = 2π . Since the function (1.89) is constant in two half-period intervals of times, it will be useful to introduce the three states (1.94) .x0 = x(0), x 21 = x(π ), x1 = x(2π ). The system (1.93) is linear and, therefore, there are matrices .F1 and .F2 relating these states as x1 = F2 x 21 . (1.95) .x 1 = F1 x0 , 2 Combining these two expressions yields x = F2 x 21 = F2 F1 x0 .

. 1

(1.96)

1 Brief Introduction to the Theory of Parametric Resonance

17

Fig. 1.5 Floquet matrix .F describes the evolution of the linearized system in a full period .T = 2π. It can be found as a product of matrices .F1 and .F2 , which describe the evolution in two half-period intervals, where the Meissner equation is autonomous

Comparing with (1.30), we recover the Floquet matrix in the form F = F2 F1 .

.

(1.97)

Expression (1.97) is very useful: the Meissner equation is autonomous in the two half-periods and, thus, the solution can be found analytically; see Fig. 1.5. Let us assume that .a > b, (1.98) and define ω1 =

.

√ √ a + b, ω2 = a − b.

(1.99)

In the interval .0 < τ < π , Eq. (1.91) becomes .

d 2ϕ + ω12 ϕ = 0, dτ 2

(1.100)

and its general solution can be written as ϕ = a1 cos ω1 τ +

.

a2 sin ω1 τ, ω1

(1.101)

with arbitrary real coefficients .a1 and .a2 . For the second variable, this yields ψ=

.

dϕ = −a1 ω1 sin ω1 τ + a2 cos ω1 τ. dτ

Expression (1.101) and (1.102) are written together as

(1.102)

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A. A. Mailybaev

⎛ ( ) ϕ cos ω1 τ .x = =⎝ ψ −ω1 sin ω1 τ

⎞ sin ω1 τ ( ) a1 ω1 ⎠ a . 2 cos ω1 τ

(1.103)

Evaluating this expression at .τ = 0 and .π , we obtain ( ) a1 , x 12 = F1 x0 , .x0 = x(0) = a2 ⎛

with

cos ω1 π .F1 = ⎝ −ω1 sin ω1 π

⎞ sin ω1 π ω1 ⎠ . cos ω1 π

(1.104)

(1.105)

Similar analysis in the second interval, .π ≤ τ ≤ 2π, yields ⎛ cos ω2 π F2 = ⎝ −ω2 sin ω2 π

.

⎞ 1 sin ω2 π ⎠ . ω2 cos ω2 π

(1.106)

Finally, the Floquet matrix is obtained as the product (1.97).

1.7.2 Stability Conditions It is easy to see from (1.105) and (1.106) that det F1 = 1, det F2 = 1.

(1.107)

det F = det (F2 F1 ) = det F1 det F2 = 1.

(1.108)

.

This means that .

Recall that the determinant of the Floquet matrix equals to the product of its eigenvalues .ρ1 and .ρ2 : . det F = ρ1 ρ2 . (1.109) At the same time, the sum of the eigenvalues equals to the trace of the matrix: tr F = ρ1 + ρ2 .

.

Also, if .ρ1 is complex, we have .ρ2 = ρ 1 .

(1.110)

1 Brief Introduction to the Theory of Parametric Resonance

19

Hence, there are three possibilities: (a) The eigenvalues are real and distinct with .ρ2 = 1/ρ1 . In this case .|tr F| > 2. By Theorem 1.2, the equilibrium of the linearized system is unstable, because .|ρ| > 1 for one of the eigenvalues. (b) The two distinct eigenvalues are complex conjugate with the unit absolute value: ±iα .ρ1,2 = e . In this case .|tr F| < 2. By Theorem 1.2, the equilibrium of the linearized system is stable. (c) The intermediate case, when the eigenvalues are mulitple (double) with the value .ρ1 = ρ2 = 1 or .ρ1 = ρ2 = −1. In this case .|tr F| = 2. Stability or instability of the linearized system depends on the Jordan structure of this double eigenvalue. Summarizing, we have the following stability criteria stability : tr F < 2,

(1.111)

instability : tr F > 2, stability boundary : tr F = 2,

(1.112) (1.113)

.

where trace is computed from the relations (1.97), (1.105) and (1.106) as ( tr F = 2 cos ω1 π cos ω2 π −

.

ω1 ω2 + ω2 ω1

) sin ω1 π sin ω2 π.

(1.114)

1.7.3 Stability Diagram Let us describe now the structure of instability regions in the parameter plane .(a, b). These parameters are related to .ω1 and .ω2 by relations (1.99). Our analysis will be based on the relation | | ) ( | | ω2 ω1 | sin ω1 π sin ω2 π || = 2, + (1.115) .tr F = 2 cos ω1 π cos ω2 π − | ω2 ω1 which according to (1.113) and (1.114) defines the boundary between stability and instability domains. When the amplitude .Δ of the support oscillation is small, the value of .b in (1.90) is also small. According to (1.99), this implies .ω1 ≈ ω2 . In particular, for √ the case .Δ = 0 (pendulum with a fixed support), we have .b = 0 and a. Then, expression (1.114) reduces to .ω1 = ω2 = ( √ ) ( √ ) ( √ ) tr F = 2 cos2 π a − 2 sin2 π a = 2 cos 2π a .

.

(1.116)

Thus, .tr F ≤ 2 and, hence, the system is stable. This is expected because the pendulum with a fixed support is stable. However, the boundary condition (1.115) in √ the case (1.116) is satisfied only when .2 a is integer. This yields a set of resonant points

20

A. A. Mailybaev

( (a, b) =

.

) n2 , 0 , n = 1, 2, . . . 4

(1.117)

These points mark the locations of possible instability regions, which may appear for nonzero but small values of .b. Observe that odd and even values of .n correspond to different types of double eigenvalues odd n :

.

ρ1 = ρ2 = −1;

even n :

ρ1 = ρ2 = 1.

(1.118)

For example, let us consider the parameters a=

.

ε2 1 ε + 2, b = , 4 π π

(1.119)

for small .ε, in which case ω1 =

.

ε ε 1 1 + , ω2 = − . 2 π 2 π

(1.120)

Computing the trace (1.114) yields ( ) 1/4 + ε2 /π 2 2 tr F = −2 sin2 ε + cos ε . 1/4 − ε2 /π 2

.

(1.121)

Since the pre-factor of .cos2 ε is greater than unity, we have .|tr F| > 2 for small positive .ε and, hence, the system is unstable. The form of instability zones near the resonant points (1.117) can be found by asymptotic methods or numerically. They have the form of “tongues” as illustrated in Fig. 1.6; the stability diagram is symmetric for positive and negative .b. For the figure demonstrating the instability zones for large values of .b we refer an interested reader to the paper [4]. Note that the boundaries of each instability zone passing through resonant points (1.117) for odd .n are characterized the double eigenvalue .ρ = −1, as described in (1.118). Conversely, the boundaries of instability zones passing through points (1.117) for even .n are characterized the double eigenvalue .ρ = 1. Since the boundaries of the adjacent zones correspond to different double eigenvalues, they cannot intersect, while boundaries of the same zone may have self-intersections. Figure 1.7 shows the change of eigenvalues with increasing .a for a fixed small value of .b.

1 Brief Introduction to the Theory of Parametric Resonance

21

Fig. 1.6 Structure of instability (grey) zones for the Meissner equation on the plane of parameters. The resonance points correspond to .a = 1/4, 1, 9/2, . . . and .b = 0

Fig. 1.7 Schematic plot of eigenvalues changing with increasing .a and fixed small value of .b

1.8 Mathieu Equation Let us considers the support of the pendulum oscillating as .

h(t) = −Δ cos Ωt.

(1.122)

The change of variables (1.87) and parameters (1.90) yields the Mathieu equation .

d 2ϕ + (a + b cos τ ) ϕ = 0. dτ 2

(1.123)

Analysis of this equation is similar to the case of the Meissner equation, except that the solution for the Floquet matrix cannot be found analytically. However, both

22

A. A. Mailybaev 2

Fig. 1.8 Instability (grey) regions for the Mathieu equation

1

0 -1

0

1

2

3

systems coincide for .b = 0 and, therefore, the resonant points (1.117) are the same. These resonance points give rise to instability regions shown in Fig. 1.8. In this figure we also showed the instability region for negative values of .a. Notice that the stability in (1.111) was not asymptotic, because both eigenvalues belong to the unit circle. Thus, this is the special case of Lyaunov, and one cannot conclude from the linear analysis that the equilibrium is stable for the original nonlinear system. One of the ways to understand the stability properties for the nonlinear system is to add a small dissipative term into the equation of motion. For the linearized system the resulting damped Mathieu equation becomes .

d 2ϕ dϕ +δ + (a + b cos τ ) ϕ = 0, dτ 2 dτ

(1.124)

where .δ > 0 is a small damping coefficient. Stability analysis of this system (or analogous system for the damped Meissner equation) follows the same ideas, with the difference that the eigenvalues do not satisfy any more the relation .ρ1 ρ2 = 1. It turns out that the eigenvalues, which were on the unit circle for the undamped system, are shifted inside the unit circle, when small damping is added; see Fig. 1.9. As a result, the stability regions increase and become the regions of asymptotic stability with both eigenvalues .|ρ| < 1 when we add a damping term; see Fig. 1.9. Accordingly, the instability zones decrease. The resulting asymptotic stability does not depend on higher-order nonlinear terms. In the limit .δ → 0, one recovers the instability zones of the undamped equation in Fig. 1.8. This argument confirms that the stability analysis using the linearized system yields the valid results.

1 Brief Introduction to the Theory of Parametric Resonance

23

Fig. 1.9 The eigenvalues, which were on the unit circle for the undamped system, are shifted inside the unit circle, when small damping is added

1.9 Physical Interpretations and Applications As one can see from Figs. 1.6 and 1.8, the largest instability zone for small .b > 0 corresponds to .a ≈ 1/4 (the first resonance point with .n = 1). Taking the value of .a from (1.90), we write the condition .a ≈ 1/4 as / Ω ≈ 2ω,

.

ω=

g , l

(1.125)

where .ω is a natural frequency of a pendulum with a fixed support. This explains why the parametric resonance is usually observed at the excitation frequencies twice larger than natural frequencies of the system.

1.9.1 Stabilization of Inverted Pendulum Substituting .g by .−g, one obtains equations of motion for the inverted pendulum. In this case, the equilibrium .ϕ = 0 is unstable if the pendulum support does not move. Instead of (1.90), we now have the parameters a=−

.

g Δ , b= . lΩ 2 l

(1.126)

Since.a < 0, these parameters correspond to the left part of the instability diagram in Fig. 1.8. It follows from the figure that, given .b > 0, the system can be stabilized if the negative value of.a is sufficiently close to zero. According to (1.126), this requires

24

A. A. Mailybaev

the excitation frequency .Ω to be sufficiently large. This explains the phenomenon of stabilization of the inverted pendulum for large excitation frequencies.

1.9.2 Ion Traps Another interesting application if the ion trap, which corresponds to a charged particle subjected to two periodic fields; see [5] for more details. The first (horizontal) field is designed such that the horizontal motion is governed by the Mathieu equation .

d2x + (a + b cos τ ) x = 0. dτ 2

(1.127)

The second (vertical) field is designed such that the vertical motion is governed by the Mathieu equation .

d2 y + (−a + b cos τ ) y = 0, dτ 2

(1.128)

with the opposite sign of the constant coefficient .a. Since the interaction with the field depends on the particle charge (not mass), the coefficients are obtained as a=

.

B A , b= , 2 mΩ m

(1.129)

where .m is the particle mass; . A and . B are some positive constants. The stability region corresponds to the intersection of the stability region for the horizontal motion (1.127) and its mirror image corresponding to the vertical motion (1.128) as shown in Fig. 1.10. The parameters (1.129) for various masses belong to the straight line, where the exact point depends on the mass of a particle under consideration. In the ion trap, the parameters . A and . B are tuned such that the line passes through the small corner of the stability region, and such that the mass of interest corresponds to a point inside this small stability region. The resulting device stabilizes particles of the specifically chosen mass at a specific point of the space. Other particles are unstable and will escape. This idea has been extensively used for ion traps: devices that allow trapping single atoms or molecules in space. In 1989, the inventors of this technology, Hans G. Dehmelt and Wolfgang Paul, were awarded a Nobel Prize in Physics.

1 Brief Introduction to the Theory of Parametric Resonance Fig. 1.10 Ion trap: the stability domain is given by the intersection of stability regions for horizontal and vertical motions (white zone). The parameters for particles of different masses belong to a straight (red dotted) line, which contains a tiny segment intersecting the stability domain. The system is designed such that the mass of a particle of interest corresponds to a point of this segment (red dot)

25

0.4

0.2

0 -0.1

-0.05

0

0.05

0.1

References 1. 2. 3. 4.

V. I. Arnol’d. Ordinary differential equations. Springer, 1992. V. I. Arnol’d. Mathematical methods of classical mechanics. Springer, 2013. P. D. Lax. Linear algebra and its applications. Wiley, 2007. A. Markeev. Stability of an equilibrium position of a pendulum with step parameters. International Journal of Non-Linear Mechanics, 73:12–17, 2015. 5. W. Paul. Electromagnetic traps for charged and neutral particles. Reviews of Modern Physics, 62(3):531, 1990. 6. A. P. Seyranian and A. A. Mailybaev. Multiparameter stability theory with mechanical applications. World Scientific, 2003.

Chapter 2

An Introduction to Parametrically Excited Systems and Their Importance on the Dynamics of Slender Structures Guilherme Rosa Franzini and Guilherme Jorge Vernizzi

Abstract This chapter brings an introduction on the dynamics of parametrically excited system. After a brief presentation of the problem in systems with one degree of freedom, some analytical and numerical methods for investigating this class of problems are presented and discussed. Since the parametric excitation is an important phenomenon on the dynamics of slenders structures, an overview of recent contributions on the theme is brought together with aspects related to its mathematical modeling.

2.1 Introduction A system is said to be parametrically excited when at least one parameter of the corresponding equations of motion explicitly depends on time. A simple but widely investigated problem that deals with parametric excitation is the simple pendulum of mass .m and length .l subjected to vertical and harmonic motion . y p (t) applied to the support; see Fig. 2.1. Particularly, if this prescribed motion is harmonic and monochromatic with the form. y p (t) = yˆ cos(Ωt) and no damping is considered, the linearized (around.q = 0) equation of motion is: .

d 2q + dt 2

(

) g − Ω 2 yˆ cos(Ωt) q = 0. l

(2.1)

G. R. Franzini (B) · G. J. Vernizzi Offshore Mechanics Laboratory, Escola Politécnica da USP (LMO/EPUSP), São Paulo, Brazil e-mail: [email protected] G. J. Vernizzi e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. R. Castilho Piqueira et al. (eds.), Lectures on Nonlinear Dynamics, Understanding Complex Systems, https://doi.org/10.1007/978-3-031-45101-0_2

27

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G. R. Franzini and G. J. Vernizzi

Fig. 2.1 Pendulum under support excitation

For the sake of generality, it is interesting to obtain a dimensionless mathematical model. For this, we consider the dimensionless quantities: τ=

.

Ω 4g 4 yˆ t, δ = 2 , 2∈ = − . 2 Ω l l

(2.2)

d ( ), the dimensionless equation of motion is given by Eq. 2.3 Defining .(˙) = dτ (undamped Mathieu’s equation).

q¨ + (δ + 2∈ cos(2τ ))q = 0.

.

(2.3)

Notice that dissipative effects can be easily included in the mathematical model. For example, if linear damping is considered, the damped Mathieu’s equation reads: q¨ + μq˙ + (δ + 2∈ cos(2τ ))q = 0.

.

(2.4)

In Eqs. 2.3 and 2.4, the parametric excitation frequency is .2 and its amplitude is associated with .∈. It is clear to the reader that different forms of the Mathieu’s equation are obtained, depending on the chosen dimensionless quantities. As a classical subject, discussions regarding the Mathieu’s equation can be found in a number of excellent textbooks such as [1, 14, 15, 17]. It is important to emphasize that the Strutt’s diagram is one of the most relevant aspects related to the Mathieu’s equation. In this diagram, regions of bounded or unbounded responses are depicted as functions of .δ and .∈, the parameters that govern the parametric excitation. In this chapter, the curves delimiting the regions of bounded or unbounded solutions (transition curves) are obtained in different forms. This chapter is structured as follows. In Sect. 2.2, the main aspects of the Floquet Theory are addressed and applications to the Mathieu’s equation are discussed. Section 2.3 brings asymptotic analyses developed with the method of multiple scales (MMS) for one degree-of-freedom systems. In Sect. 2.4, the harmonic balance method (HBM) is employed for determining the transitions curves associated with the Mathieu’s equation. Finally, Sect. 2.6 presents a literature review focusing on the importance of parametric excitation in the dynamics of slender structures.

2 An Introduction to Parametrically Excited Systems and …

29

2.2 Floquet Theory and Application to the Mathieu’s Equation Floquet Theory is an approach for studying the stability of periodic orbits for both autonomous and non-autonomous systems. Section 2.2.1 brings an overview about this theory and is based on the books [16, 17]. An application of the Floquet Theory to the Mathieu’s equation is described in Sect. 2.2.2.

2.2.1 An Overview on the Floquet Theory Firstly, we introduce the Floquet Theory for an autonomous systems. A system of first-order ordinary differential equations is said to be autonomous if it is written in the form of Eq. 2.5, where .n is the dimension of the phase-space, .xi , i = 1, 2, . . . , n are the state-variables and.μ is the vector that collects the parameters of the mathematical model. In the case of a mechanical system, .μ gathers, for example, the mass, the stiffness and the damping. Notice that time does not explicitly appear in the governing equations for an autonomous system. ⎧ ⎫ ⎧ x˙1 (t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ x˙2 (t) ⎪ ⎬ ⎪ (t) x ˙ 3 = . ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ⎭ ⎪ x˙n (t)

⎫ f 1 (x1 (t), x2 (t), x3 (t), . . . , xn (t), μ) ⎪ ⎪ ⎪ f 2 (x1 (t), x2 (t), x3 (t), . . . , xn (t), μ) ⎪ ⎪ ⎬ f 3 (x1 (t), x2 (t), x3 (t), . . . , xn (t), μ) → x˙ (t) = F(x(t), μ). ⎪ .. ⎪ ⎪ . ⎪ ⎪ ⎭ f n (x1 (t), x2 (t), x3 (t), . . . , xn (t), μ) (2.5) Let . X 0 = X 0 (t) being a periodic solution of Eq. 2.5 with minimal period .T at .μ = μ0 . Consider a disturbed solution . x = x(t) obtained by summing . y = y(t) to . X 0 . The linearization of Eq. 2.5 around . X 0 allows obtaining Eq. 2.6. .

˙y(t) ≈ A(t, μ0 ) y(t),

(2.6)

with . A(t, μ0 ) = D X F(X 0 , μ0 ) being the Jacobian matrix. Since Eq. 2.6 is linear, it has.n linearly independent solution. yi (t), i = 1, 2, . . . , n. Hence, any solution of Eq. 2.6 is obtained by linear combination of these fundamental solutions, collected as the columns of matrix.Y (t). From the latter definition, we write Eq. 2.7. .

[ ] Y (t) = y1 (t) y2 (t) y3 (t) . . . yn (t) → Y˙ (t) = A(t, M0 )Y (t).

(2.7)

Now, we demonstrate that functions with the form . yi (t + T ), i = 1, 2 . . . , n also correspond to a set of fundamental solutions. For this, consider the change of variables .τ = t + T in Eq. 2.7 and notice that . A has period .T . These observations lead to Eq. 2.8, which allows concluding that the columns of .Y (t + T ) are,

30

G. R. Franzini and G. J. Vernizzi

indeed, fundamental solutions of Eq. 2.6 and can be written as a linear combination of . yi (t), i = 1, 2 . . . , n (see Eq. 2.9). .

dY (τ − T ) = A(τ − T )Y (τ − T ) = A(τ )Y (τ ), dτ

(2.8)

Y (t + T ) = Y (t)Φ.

(2.9)

.

In Eq. 2.9, we do not have any information about the coefficients of the .n × n matrix .Φ. Notice, however, that .Φ maps the dynamic behavior from the instant .t to the instant .t + T . If we conveniently choose .Y (0) = I, we have .Φ = Y (T ). It is worth mentioning that .Φ is named monodromy matrix. The eigenvalues of .Φ (.ρi , i = 1, . . . , n) are the Floquet multipliers and play an important role on the stability of periodic orbits. As any fundamental solution of Eq. 2.6 is a linear combination of the fundamental solutions . yi (t + T ), i = 1, 2 . . . , n, a different fundamental matrix . V (t) satisfies −1 .Y (t) = V (t) P , . P being a .n × n non-singular constant matrix. Using the latter definition and Eq. 2.9, the following relations hold. .

Y (t + T ) = Y (t)Φ ↔ V (t + T ) P −1 = V (t) P −1 Φ ↔ V (t + T ) = V (t) P −1 Φ P = V (t) J.

(2.10)

We can conveniently choose . P (and, consequently, . J). Firstly, consider the case in which the eigenvalues of .Φ are distinct. In this case and denoting the eigenvectors of .Φ as . pi (.i = 1, . . . , n), the choice of . P = [ p1 p2 . . . pn ] yields .

J = P −1 Φ P = P −1 Φ[ p1 p2 . . . pn ] = = P −1 [Φ p1 Φ p2 . . . Φ pn ] = P −1 [ρ1 p1 ρ2 p2 . . . ρn pn ] = ⎤ ⎡ ρ1 0 . . . 0 ⎢ 0 ρ2 . . . 0 ⎥ ⎥ ⎢ = P −1 P D = D = ⎢ . . . . ⎥ . ⎣ .. .. . . .. ⎦

(2.11)

0 0 . . . ρn

From Eqs. 2.10 and 2.11, the solution .v m (t) after one period .T satisfies Eq. 2.12, from which one clearly notices that the Floquet multipliers define a qualitative aspect of .v. If .|ρm | > 1, .vm (t) → ∞ if . N (and, consequently, .t) goes to infinity. On the other hand .vm (t) → 0 when . N → ∞ if .|ρm | < 1. v (t + T ) = ρm v m (t), m = 1, 2, . . . , n ↔

. m

↔ v m (t + N T ) = ρmN v m (t), m = 1, 2, . . . , n and N integers.

(2.12)

2 An Introduction to Parametrically Excited Systems and …

31

Also from Eq. 2.12 and considering the case.ρm = 1,.v m (t + T ) = v m (t), defining a periodic solution of period .T . In turn, .ρm = −1 leads to .v m (t + T ) = −v m (t) and .v m (t + 2T ) = v m (t) (periodic solution of period .2T ). At this point, an important nomenclature is introduced. Multiplying both sides of the identity .vm (t + T ) = ρm v(t) by .e−γm (t+T ) , we obtain v (t + T )e−γm (t+T ) = ρm e−γm t e−γm T vm (t).

. m

If .γm is defined as .γm =

1 T

(2.13)

ln ρm Eq. 2.13 becomes

v (t + T )e−γm (t+T ) = vm (t)e−γm t .

. m

(2.14)

From Eq. 2.14, one clearly notices that.φm (t) = vm (t)e−γm t is periodic with period T . The Floquet or normal form of the solution is.vm (t) = φm (t)eγm t and the quantities .γm are known as characteristic exponents. We discuss an aspect valid for autonomous systems. The derivative of Eq. 2.5 with respect to time yields: ¨ = D X F(x, M) x˙ . .x (2.15) .

If . x is a solution of Eq. 2.5, . x˙ is solution of both Eqs. 2.15 and 2.6. Since . X 0 (t) is solution of Eq. 2.5, . X˙0 (t) is solution of Eq. 2.6 and has period .T . Hence . X˙0 (t) = X˙0 (t + T ) and, particularly,. X˙0 (0) = X˙0 (T ). Since . X˙0 (t) is a solution of Eq. 2.6, it can be written as a linear combination of . y1 (t), y2 (t), . . . , yn (t) as: ˙0 (t) = Y (t)α, .X (2.16) α being a vector of constants. From Eq. 2.16,. X˙0 (0) = Y (0)α and. X˙0 (T ) = Y (T )α. Recalling that . X˙0 (0) = X˙0 (T ), .Y (0) = I and .Y (T ) = Φ, we have:

.

Φα = α.

.

(2.17)

Equation 2.17 indicates that one eigenvalue of the monodromy matrix associated with an autonomous system is .1. As seen in this section, the Floquet multipliers determine the stability of the periodic solutions.1 As pointed out in [16] in the discussion of autonomous systems, a periodic solution of Eq. 2.5 is hyperbolic if only one Floquet multiplier is on the unit circle defined in the complex plane. The hyperbolic solution is asymptotically stable if no Floquet multiplier is located outside the unit circle and unstable if at least one Floquet multiplier is outside the unit circle and corresponds to an unstable limit cycle.

1

In studies using the indirect Lyapunov’s method, the real parts of the eigenvalues of the Jacobian matrix determine the stability of equilibrium of an autonomous system.

32

G. R. Franzini and G. J. Vernizzi

For a non-autonomous system (i.e., governed by. x˙ (t) = F(x(t), μ, t), most of the above comments remain valid. Notice, however, that the presence of at least one Floquet multiplier equal to .1 is only valid for autonomous systems. For non-autonomous system, this property is no longer guaranteed. In this scenario, a hyperbolic periodic solution of a non-autonomous system is the one in which none of the Floquet multipliers are located on the unit circle. Now, we briefly comment the case in which the eigenvalues are not distinct. In this case, the number of independent eigenvectors may not be sufficient to define2 a basis and matrix. J can be either diagonal or composed of Jordan’s blocks. A Jordan’s block has the repeated eigenvalue in the main diagonal, .1 in the upper diagonal and .0 in the other positions. The reader interested in details regarding the Floquet Theory when the eigenvalues are not distinct is refereed to [17].

2.2.2 Application to the Mathieu’s Equation Firstly, we discuss a numerical application of the Floquet Theory to the damped Mathieu’s equation (Eq. 2.4). The focus is to identify regions of the plane of control parameters .(δ; ∈) (the parameters that govern the parametric excitation) associated with bounded or unbounded responses. For this, consider the following steps: 1. Discretization of the plane .(δ; ∈) using a certain grid; 2. Write the mathematical model in the form of a first-order system of differential equations . x˙ (t) = f (x(t), μ, t). In this case, .μ gathers the amplitude and frequency of the parametric excitation (.∈ and .δ, respectively) and the damping .μ; 3. For each point of the discrete domain, numerically integrate the mathematical model during one period .T using as non-trivial initial condition .xk (0), .k = 1, 2, . . . , n, .n being the dimension of the first-order system. For each simulated initial condition, the state-vector . x(T ) defines the .kth-column of the monodromy matrix; 4. Once the monodromy matrix is obtained, calculate its eigenvalues (Floquet multipliers); 5. For the considered pair .(δ; ∈), assign .ρ∗ = max{|ρk |}, .k = 1, 2, . . . , n; 6. At the end of step 5, we obtain .ρ∗ (δ; ∈), which can be plotted in the form of a colormap. Regions characterized by .ρ∗ > 1 are associated with unbounded responses. For the damped Mathieu’s equation (Eq. 2.4), it is clear that the parametric excitation period is.T = π. Figure 2.2 presents plots.ρ∗ (δ, ∈) for different values of damping .μ of the Mathieu’s equation. Each plot shown in Fig. 2.2 has been obtained using 2

Two important concepts are recalled. The algebraic multiplicity is the number of times that a repeated eigenvalue appears. The geometric multiplicity is the number of independent eigenvectors associated with a certain repeated eigenvalue. If the algebraic multiplicity is larger than the corresponding geometric multiplicity, we do not have sufficient eigenvectors for defining a basis and the concept of associated vector (or generalized eigenvector) must be employed.

2 An Introduction to Parametrically Excited Systems and …

33

(a) µ = 0.

(b) µ = 0.08.

(c) µ = 0.16.

(d) µ = 0.24.

Fig. 2.2 Maps .ρ∗ (δ, ∈) for different values of damping .μ

the DifferentialEquations.jl Julia package with a 2,000.×2,000 grid. In a standard notebook (i7 10th generation, 8Gb RAM), the simulation time for each map is close to .18 min. A first aspect to be noticed in Fig. 2.2, a number of regions characterized by.ρ∗ > 1 can be identified. Particularly, we define as the principal instability region the one arisen around .δ = 1. From the physical point of view, this indicates that the case in which the parametric excitation frequency is twice the natural frequency3 consists on a favorable scenario for the parametric instability. Figure 2.2 also reveals that the increase in damping shrinks the area of the plane .(δ, ∈) associated with unbounded solutions. In addition, we notice that the origin of the transition curves moves from the abscissa to positive values of .∈. In Sects. 2.3 and 2.4, analytical derivations of the transition curves are compared with the map .ρ∗ (δ, ∈) for the undamped Mathieu’s equation. Now, we bring some analytical considerations regarding the application of the Floquet Theory to the undamped Mathieu’s equation (Eq. 2.3). Defining .x1 = q and

3

If the system has more than one degree of freedom, favorable scenarios for the parametric instability are those in which the parametric excitation frequency is twice the natural frequencies.

34

G. R. Franzini and G. J. Vernizzi

x = q, ˙ the undamped Mathieu’s equation is written as the first-order system of differential equations given by Eqs. 2.18 and 2.19.

. 2

x˙ = x2 , x˙2 = −(δ + 2∈ cos(2τ ))x1 .

. 1

(2.18) (2.19)

We denote . x I = {x I,1 x I,2 }T the solution of Eqs. 2.18 and 2.19 obtained with initial conditions.x I,1 (0) = 1 and.x I,2 (0) = 0. Similarly, we define. x I I = {x I I,1 x I I,2 }T as a solution of the undamped Mathieu’s equation with.x I I,1 (0) = 0 and.x I I,2 (0) = 1. The definition of the monodromy matrix leads to: [ Φ=

.

] x I,1 (T ) x I I,1 (T ) . x I,2 (T ) x I I,2 (T )

(2.20)

As the Floquet multipliers .ρ are the eigenvalues of the monodromy matrix, they satisfy Eq. 2.21. 2 .ρ − ρ (x I,1 (T ) + x I I,1 (T )) +Δ(T ) = 0. (2.21) ~ ~~ ~ 2η .Δ(T ) = x I,1 (T )x I I,2 (T ) − x I,2 (T )x I I,1 (T ) being the determinant of .Φ (Wronskian at .τ = T ). It is clear that the Wronskian at any instant .τ is .Δ(τ ) = x I,1 x I I,2 − x I,2 x I I,1 and, consequently

.

Δ˙ = x˙ I,1 x I I,2 + x I,1 x˙ I I,2 − x˙ I,2 x I I,1 − x I,2 x˙ I I,1 .

(2.22)

As Eqs. 2.18 and 2.19 hold for . x I and . x I I , after some algebraic work with these equations, one obtains .Δ˙ = 0. As the Wronskian does not depend on time, .Δ(T ) = Δ(0) √ = 1. Using the latter result in Eq. 2.21, the Floquet multipliers are .ρ1,2 = η ± η 2 − 1 with .ρ1 ρ2 = 1. Some conclusions regarding the presence of stable or unstable responses can be drawn by inspecting .η. If .|η| > 1, the Floquet multipliers are real, distinct and one of them has modulus larger than .1, leading to unbounded responses. On the other hand, in the case .|η | < 1, the Floquet multipliers are complex conjugate and characterized by .|ρ1 | = |ρ2 | = 1. It is clear that the transition bounded/unbounded responses is associated with .|η| = 1. If .η = 1, .ρ1 = ρ2 = 1 and at least one normal form has period . T . In turn, .η = −1 leads to .ρ1 = ρ2 = −1 and at least one normal form has period .2T .

2.3 Asymptotic Analyses Using MMS This section discusses the response of parametrically excited one-degree-of-freedom systems using MMS (see, for example, the classical textbooks [1, 17]). Section 2.3.1 focuses on obtaining the transition curves associated with the principal parametric

2 An Introduction to Parametrically Excited Systems and …

35

instability for the undamped Mathieu’s equation. On the other hand, Sect. 2.3.2 discusses applications of this method for a class of nonlinear systems derived from the Mathieu’s equation.

2.3.1 Transition Curves of the Undamped Mathieu’s Equation In this subsection, we follow the derivation made in [15]. We seek for an approximated solution of Eq. 2.3 in the form of power series in .∈, as given by Eq. 2.23. Notice, in Eq. 2.23, that unknown functions .qk , k = 0, 1, 2 depend on three distinct time scales k . Tk = ∈ τ , k = 0, 1, 2. q(τ ) = q0 (T0 , T1 , T2 ) + ∈q1 (T0 , T1 , T2 ) + ∈2 q2 (T0 , T1 , T2 ).

.

(2.23)

∂ It is convenient to define a family of differential operators as . Dm ( ) = ∂T p ( ). m . τ can be easily obtained by Using this definition, the derivatives with respect to using the chain rule. Within this framework, Eqs. 2.24–2.26 are exact up to .O(∈2 ). p

∂( ) dT1 ∂( ) dT2 ∂( ) dT0 + + = (D0 + ∈D1 + ∈2 D2 )( ), (˙) = ∂T0 dτ ∂T1 dτ ∂T2 dτ (¨) = (D02 + ∈(2D1 D0 ) + ∈2 (2D2 D0 + D12 ))( ),

.

q¨ =

D02 q0

+ ∈(2D1 D0 q0 +

D02 q1 )

+∈

2

(D02 q2

+

D12 q0

p

(2.24) (2.25)

+ 2D0 D2 q0 + 2D0 D1 q1 ). (2.26)

The focus herein lies on investigating the transition curves around the principal Mathieu’s instability, i.e., around the case with .δ = 1. Hence, we expand .δ as .δ = 1 + ∈δ1 + ∈2 δ2 . The method follows by rewriting Eq. 2.3 using both the expansion for .δ and the relations defined by Eqs. 2.23–2.26 and, then, collecting the terms of equal power in .∈. This leads to: O(1) : D02 q0 + q0 = 0,

(2.27)

O(∈) : D02 q1 + q1 = −(ei2T0 + e−i2T0 )q0 − δ1 q0 − 2D1 D0 q0 , O(∈2 ) : D02 q2 + q2 = −δ2 q0 − (ei2T0 + e−i2T0 )q1 − δ1 q1 − − D12 q0 − 2D2 D0 q0 − 2D1 D0 q1 .

(2.28)

.

(2.29)

It is straightforward to see that the solution of Eq. 2.27 is .q0 = Aei T0 + A∗ e−i T0 = Ae + c.c., where . A = A(T1 , T2 ) is complex amplitude slow varying with respect to the fast time scale .T0 , . A∗ corresponds to its complex conjugate and .c.c. refers to the complex conjugate of the expression immediately at left. By substituting .q0 into the right-hand side of Eqs. 2.28, 2.30 is obtained after some algebraic manipulations. i T0

36

G. R. Franzini and G. J. Vernizzi .

D02 q1 + q1 = −Aei3T0 − ei T0 (δ1 A − A∗ − i2D1 A) + c.c. .

(2.30)

The term multiplying .ei T0 on the right-hand side of Eq. 2.30 is called secular, since it causes unbounded responses .q1 . Hence, for the convergence of expansion given by Eq. 2.23, the secular term must be null (solvability condition). By expanding . A into its real and imaginary part as . A = Ar + i Ai , the solvability condition is given, in matrix form, by Eq. 2.31. { .

D1 A i D1 A r

}

[ =

1 (1 + δ1 ) 0 2 1 (1 − δ ) 0 1 2

]{

Ai Ar

} .

(2.31)

ˆ λt , where .λ is, at this point, Equation has (2.31) solution of the form . D1 A = Ae unknown. Using this solution in Eq. 2.31, one obtains: ([ .

] ) 1 (1 + δ1 ) 0 2 ˆ = 0. − λI A 1 (1 − δ1 ) 0 2

(2.32)

Equation 2.32 has no trivial solution if the determinant of the coefficient matrix. Mathematically, the latter condition reads: / 1 1 1 − δ12 , λ2 = −λ1 . (2.33) λ2 − (1 − δ12 ) = 0 → λ1 = 4 2 √ 2 1−δ For the sake of convenience, we define .β = 1+δ1 1 as an auxiliary quantity. Since .λ1 and .λ2 are the eigenvalues of the coefficient matrix of Eq. 2.31, the corresponding eigenvectors are .v 1 = {1 β}T and .v 2 = {1 − β}T , respectively. Hence, the solution of Eq. 2.31 reads: .

{ .

Ai (T1 , T2 ) Ar (T1 , T2 )

}

= a1 (T2 )v 1 eλ1 T1 + a2 (T2 )v 2 e−λ1 T1 .

(2.34)

Note that Eq. 2.34 is equivalent to: .

Ai (T1 , T2 ) = a1 (T2 )eλ1 T1 + a2 (T2 )e−λ1 T1 , Ar (T1 , T2 ) = a1 (T2 )βe

λ1 T1

− a2 (T2 )βe

−λ1 T1

(2.35) .

(2.36)

Equations 2.35 and 2.36 reveal that the behavior of . A(T1 , T2 ) depends on the values of .λ1 and .λ2 , both dependent on .δ1 . If .|δ1 | < 1, .λ1 is a real number larger than .0 and unbounded solutions appear. On the other hand, if .|δ1 | > 1, both .λ1 and .λ2 are complex quantities with null real part. In the latter scenario, . A(T1 , T2 ) is bounded and oscillatory. Obviously, the transition between bounded and unbounded solutions corresponds to .|δ1 | = 1. Hence, a first approximation for the transition curves is given by .δ = 1 + ∈δ1 = 1 ± ∈.

2 An Introduction to Parametrically Excited Systems and …

37

After removing the secular term in Eq. 2.30, we need to obtain the corresponding particular4 solution. It is easy to notice that this particular solution is: q =

. 1

1 i3T0 Ae + c.c. . 8

(2.37)

with . A = Ar + i Ai from Eqs. 2.35 and 2.36. Now, we are interested in obtaining a higher-order approximation for the transition curves. For this, we consider the solution for .q0 and for .q1 (the latter in Eq. 2.37) in Eq. 2.29. Using this approach, one obtains: .

D02 q2 + q2 = −

) ] [( 1 A + D12 A + i2D2 A ei T0 + c.c. + N ST, δ2 + 8

(2.38)

where . N ST indicates non-secular terms. Secular terms are removed from the solution of Eq. 2.38 if: .

( ) 1 δ2 + A + D12 A + i2D2 A = 0. 8

(2.39)

By differentiating both sides of Eqs. 2.35 and 2.36 with respect to .T1 and recalling that . A = Ar + i Ai , one easily obtains that . D12 A = λ21 A. Defining .γ = δ2 + 18 + λ21 and separating the real and the imaginary parts of the complex amplitude . A, Eq. 2.39 is rewritten as: ]{ } { } [ Ai D2 A i 0 γ2 . = (2.40) . − γ2 0 D2 A r Ar Similarly to already discussed for Eq. 2.31, the eigenvalues of coefficient matrix provide information regarding the bounded or unbounded aspect of the solution. The transition from bounded to unbounded solutions are obtained with .γ = 0. Since 1 2 2 .λ1 = (1 − δ1 ) and recalling that .δ1 = ±1 at the transition curves, the condition 4 .γ = 0 holds if: 1 1 1 − (1 − δ12 ) = − . .δ2 = − (2.41) 8 4 8 Using the above result, we obtain another approximation (correct up to quadratic terms in .∈) for the transition curves as 1 δ = 1 ± ∈ − ∈2 . 8

.

(2.42)

Figure 2.3 presents the approximations for the transition curves discussed in this section. Since the method of multiple scales is developed around the principal Mathieu’s instability (i.e., .δ = 1), it is natural that the other instability regions are not 4

The homogeneous solution of this equation is already included in the solution of the equation of order .1.

38 Fig. 2.3 Transition curves obtained with the method of multiple scales

G. R. Franzini and G. J. Vernizzi 6 5 4 3 2 1 0

0

0.5

1

1.5

2

2.5

3

3.5

Fig. 2.4 Transition curves (in black) superimposed onto the .ρ∗ (δ, ∈) map

shown. Figure 2.4 depicts the transition curves .δ = 1 ± ∈ − 81 ∈2 superimposed onto the stability map obtained using the Floquet theory. This plot clearly reveals the excellent agreement between the transition curves and the boundary of stability if .∈ < 0.7.

2.3.2 Post-critical Amplitude of a Class of Nonlinear Mathieu’s Equation Consider now a modified form of Eq. 2.3, in which a nonlinear restoration term of the Duffing form and a nonlinear damping of the Morison form are included. The Duffing–Mathieu–Morison oscillator obtained is represented by Eq. 2.43. This class

2 An Introduction to Parametrically Excited Systems and …

39

of oscillator can represent, for example, the parametric excitation of vertical slender structures immerse in fluid, as shown in Sect. 2.6. q¨ + (δ + 2∈ cos (2τ )) q + aq 3 + b |q| ˙ q˙ = 0.

(2.43)

.

Due to the presence of nonlinearities, the response of the system is bounded for all .δ and .∈, achieving a steady-state regime. The post-critical amplitude of response can then be obtained by numerical simulations of the oscillator, or be predicted by means of analytical results obtained with MMS. Following the discussions made in Sect. 2.3.1, an expansion is proposed for .q in the form of Eq. 2.44. q (τ ) = q0 (T0 , T1 ) + ∈q1 (T0 , T1 ) .

(2.44)

.

For the proper substitution of time derivatives, Eqs. 2.24–2.26 are used, but only up to order .∈ terms. The limitation of the order in the present analysis is made due to the presence of the absolute value function in Eq. 2.43, which creates a barrier to expansions in further orders. To investigate the behavior around the main parametric resonance, the expansion .δ = 1 + ∈δ1 is also made, together with the scaling .a = ∈α and .b = ∈β. Now, substituting all terms in Eq. 2.43 and collecting terms of the same order leads to the system: O(1) : D02 q0 + q0 = 0,

.

O(∈) : −

αq03

D02 q1

(

+ q1 = −2D0 D1 q0 − δ1 q0 − e

− β |D0 q0 | D0 q0 .

i2T0

+e

−i2T0

)

(2.45) q0 (2.46)

As in the linear case, the .O(1) solution gives .q0 = Aei T0 + A∗ e−i T0 = Aei T0 + c.c., with . A = A(T1 ) in the present case. Now, in order to deal with the nonlinear damping, the same strategy proposed in [17] is adopted. For that, the nonlinear damping is expanded in its frequency components using the a Fourier series decomposition, so it can be written as the sum of a secular term with non-secular ones. Equation 2.46 then becomes: ( ) D02 q1 + q1 = −2i D1 A − δ1 A − A∗ − 3α A2 A∗ − f 1 ei T0 + c.c. + N ST. (2.47) Where . f 1 is the secular component from the nonlinear damping. Now, using the polar representation in real functions . A = Beiθ , which also implies in .q0 = 2B cos (T0 + θ), with . B = B(T1 ) and .θ = θ(T1 ). With these definitions, the term . f 1 can be calculated as: ( 2π β D0 q0 |D0 q0 | e−i T0 dT0 . f1 = 2π 0 ( 2π 16iβ 2 β −2B sin (T0 + θ) |2B sin (T0 + θ)| e−i(T0 +θ) dT0 = eiθ = eiθ B . 2π 0 3π (2.48) .

40

G. R. Franzini and G. J. Vernizzi

While the condition of nullifying secular terms results in: 2i (D1 B + i B D1 θ) eiθ + δ1 Beiθ + Be−iθ + 3αB 3 eiθ +

.

16iβ 2 iθ B e = 0. (2.49) 3π

Now, dividing Eq. 2.49 by .eiθ and separating it in real and imaginary parts, the solvability condition becomes the system: 16β 2 B − B sin (2θ) = 0, 3π − 2B D1 θ + δ1 B + 3αB 3 + B cos (2θ) = 0.

2D1 B +

.

(2.50) (2.51)

Since the interest is on the steady-state condition, the derivatives. D1 are considered null. Then, isolating the trigonometric term in Eqs. 2.50 and 2.51, squaring both equations and adding them together results in: ( .

16β 3π

)2 B 2 + δ12 + (3α)2 B 4 + 6αδ1 B 2 = 1.

(2.52)

The solution is then given by: ( − 6αδ1 + .

B2 =

(

16β 3π

)2 )

/( +

6αδ1 +

(

2 (3α)2

16β 3π

)2 )2

( ) − 4 (3α)2 δ12 − 1 . (2.53)

The solution can be rewritten with some algebraic work and using.δ1 = (δ − 1)/∈, leading to:

.

B 2=

/( ( ( )2 )2 ( ) ( )2 ) 6a(δ−1)+ 16b + −4(3a)2 (δ−1)2−∈2 − 6a(δ−1)+ 16b 3π 3π 2 (3a)2

. (2.54)

Notice that the condition between .0 or finite amplitude solutions is given by δ = 1 ± ∈ which is the same as presented in Eq. 2.42. This is given by the condition that the left-hand side of Eq. 2.54 is positive. It is also important to emphasize that the amplitude of motion is given by .A = 2B, considering the polar decomposition made. In order to address the applicability of the obtained solution, it is compared to the amplitudes obtained from numerical simulations of the system represented by Eq. 2.43. Considering .a = 0.02 and .b = 0.3, simulations were performed with .0 ≤ δ ≤ 2 and .0 ≤ ∈ ≤ 2 using a 1,000.×1,000 grid, with a total dimensionless time .τ = 600. The amplitudes are then obtained from the steady-state response, taken from the final .10% of the simulations. The numerical integration is made using the native Matlab® ode45 function in a standard desktop (i7 7th generation, 16Gb RAM), taking roughly 4 h for the entire grid. The results are shown in Fig. 2.5. In turn, the .

2 An Introduction to Parametrically Excited Systems and …

41

Fig. 2.5 Post-critical amplitude map obtained with numerical integration

Fig. 2.6 Post-critical amplitude map obtained with an analytical solution using the method of multiple scales

map can be made with the analytical solution, in the same computer and with same grid, in a fraction of a second, being possible to say it takes the time of the click of a button. This second map is shown in Fig. 2.6. As it can be seen by comparing Figs. 2.5 and 2.6, there is a very good agreement between the numerically obtained amplitudes and those from the derived analytical solution. In order to complement the comparison and show the adherence of the analytical solution with the numerical integration of the system, a comparison between the obtained amplitudes made for the particular case .δ = 1 is shown in Fig. 2.7.

42

G. R. Franzini and G. J. Vernizzi

Fig. 2.7 Comparison between numerical and analytical predictions for the post-critical amplitude for .δ = 1

8 Numerical Analytical

6

4

2

0

0

0.5

1

1.5

2

2.4 Use of HBM The section is inspired by the discussion made in the lectures [21]. As discussed in Sect. 2.2, the transition curves for the undamped Mathieu’s equation are associated with periodic solutions of period either .T = π or .T = 2π. Hence, the solutions in the transition curves can be written in the form of Fourier Series given by Eq. 2.55. ∞ ( ∞ ( ) ∑ ) ∑ ˜ .q(t) = a˜ n cos n2τ + bn sin n2τ + c˜n cos nτ + d˜n sin nτ . n=0

(2.55)

n=0

Notice that the even harmonics (i.e., those with the form.2nτ , n = 1, 2, . . .) appear in the two sums. Hence, we can write just one sum: q(t) =

∞ ∑

.

(an cos nτ + bn sin nτ ) .

(2.56)

n=0

The following identities are useful: (

) ( inτ ) ei2τ + e−i2τ e + e−inτ . cos 2τ cos nτ = = 2 2 1 1 = cos((2 + n)τ ) + cos((2 − n)τ ), 2 2 ( i2τ ) ( inτ ) e + e−i2τ e − e−inτ cos 2τ sin nτ = = 2 2i 1 1 = sin((2 + n)τ ) + sin((n − 2)τ ), 2 2

(2.57) (2.58)

2 An Introduction to Parametrically Excited Systems and …

43

q cos 2τ = ) )] ( ∞ [ ( ∑ cos((2+n))τ +cos((2−n))τ sin((2+n))τ +sin((n−2))τ = +bn , an 2 2 n=0 (2.59) q¨ =

∞ ∑

−n 2 (an cos nτ + bn sin nτ ) .

(2.60)

n=0

Aiming at illustrating the method, we firstly consider an expansion considering three harmonics. In this scenario, we have: δq = δa0 + δa1 cos τ + δb1 sin τ + δa2 cos 2τ + δb2 sin 2τ a0 a1 b1 q cos 2τ = (cos 2τ + cos 2τ ) + (cos 3τ + cos τ ) + (sin 3τ − sin τ )+ 2 2 2 b2 a2 sin 4τ , + (cos 4τ + 1) + 2 2 q¨ = −(a1 cos τ + b1 sin τ + 4a2 cos 2τ + 4b2 sin 2τ ). (2.61)

.

By substituting Eq. 2.61 into the Mathieu’s equation and collecting the terms of the same trigonometric function (harmonic balance), we obtain: (−a1 +δa1 +∈a1 ) cos τ +(−b1 +δb1 −∈b1 ) sin τ +(−4a2 +δa2 +2∈a0 ) cos 2τ+ + (−4b2 + δb2 ) sin 2τ + (δa0 + ∈a2 ) + . . . = 0, (2.62)

.

where (.. . .) represents harmonic of higher-order, not considered in the expansion. Due to the orthogonality between the trigonometric functions, Eq. 2.62 holds if each term inside brackets vanishes. This condition can be written in matrix form as ⎛ ⎞⎧ ⎫ ⎧ ⎫ 0 −1 + δ + ∈ 0 0 0 0⎪ ⎪ ⎪ ⎪ ⎪ ⎪a0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎜0 ⎪ ⎟⎪ ⎪ ⎪ a 0 −1 + δ − ∈ 0 0 0⎪ ⎬ ⎨ ⎨ ⎬ 1 ⎜ ⎟ ⎟ b1 = 0 . 0 0 −4 + δ 0 . ⎜2∈ (2.63) ⎜ ⎟⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎝0 a 0 0 0 δ − 4⎠ ⎪ 0 ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎭ ⎪ ⎩ ⎪ ⎩ ⎪ ⎭ b2 δ 0 0 ∈ 0 0 Non-trivial solutions of Eq. 2.63 exist if | | | 0 −1 + δ + ∈ 0 0 0 || | |0 0 −1 + δ − ∈ 0 0 || | 0 0 −4 + δ 0 || = . |2∈ | |0 0 0 0 δ − 4|| | |δ 0 0 ∈ 0 | = (4δ − δ 2 + 2∈2 )(−4 + δ)(−1 + δ − ∈)(−1 + δ + ∈) = 0.

(2.64)

44

G. R. Franzini and G. J. Vernizzi

Fig. 2.8 Transition curves (in magenta) superimposed onto the .ρ∗ (δ, ∈) map

The transition curves are defined by imposing each term inside brackets in Eq. 2.64 equal to zero. Particularly, the last two terms lead to .δ = 1 + ∈ and .δ = 1 − ∈, the same results obtained from the MMS as a first approximation. The remaining terms in Eq. 2.64 define other transition curves. The increase in the number of terms considered in HBM improves the representation of the transition curves but highly increase the algebraic work. In this scenario, symbolic computation is mandatory. Considering.7 harmonics in HBM, the following approximations for the transition curves are obtained: 2304 − 784δ + 56δ 2 − δ 3 − 40∈2 + 2δ∈2 = 0,

(2.65)

.

225 − 259δ + 35δ − δ + 225∈ − 34δ∈ + δ ∈ − 26∈ + 2δ∈ − ∈ = 0, (2.66) 2

3

2

2

2

3

2304δ − 784δ 2 + 56δ 3 − δ 4 + 1152∈2 − 144δ∈2 + 4δ 2 ∈2 − 2∈4 = 0,

(2.67)

− 11025 + 12916δ − 1974δ + 84δ − δ + 11025∈ − 1891δ∈+ 2

3

4

+ 83δ 2 ∈ − δ 3 ∈ + 1283∈2 − 134δ∈2 + 3δ 2 ∈2 − 58∈3 + 2δ∈3 − ∈4 = 0.

(2.68)

Figure 2.8 shows the transition curves obtained with HBM using seven harmonic superimposed onto the .ρ∗ (δ, ∈) map. This plot clearly reveals that the solutions from HBM agree with the qualitative aspects inferred from the .ρ∗ (δ, ∈) map.

2.5 Bifurcation of Periodic Orbits As discussed in Sect. 2.2, the transitions from bounded to unbounded solutions are associated with the presence of some Floquet multipliers with modulus equal to .1. Hence, bifurcations of periodic orbits are characterized by the escape from the unit circle of at least one Floquet multiplier as one (or more) parameters of the mathematical model is changed.

2 An Introduction to Parametrically Excited Systems and …

45

6 5

1

4

0.5

3

0

2

-0.5

1

-1

0 0.5

1

1.5

2

2.5

3

3.5

-2

4

(a) Investigated region.

-1

0

1

2

(b) Examples of Floquet multipliers. 5

1.5 1

4

0.5

3

0 2

-0.5

1

-1 -1.5 -2

-1

0

1

2

0

(c) Investigated region. Fig. 2.9 Bifurcation of periodic orbits for the undamped Mathieu’s equation

Following [16], when one Floquet multiplier leaves the unit circle through .−1, a period-doubling (or flip) bifurcation is observed. If the the Floquet multiplier leaves the mentioned circle through .+1, one of the following types of bifurcations can be observed: symmetry-breaking, cyclic-fold and transcritical. Finally, the NeimarkSacker bifurcation is characterized by the two complex conjugate Floquet multipliers leaving the unit circle. Aiming at illustrating the bifurcations of periodic orbits observed for the undamped Mathieu’s equation, consider the transition curves in the vicinity of the principal Mathieu’s instability obtained with HBM and shown in Fig. 2.9a. We follow the Floquet multipliers as.δ is varied keeping.∈ = 1.5 (see red dashed line). The intersections between the red dashed line and the transition curves occur at.δ = 2.17 and.δ = 3.81. In this same figure, the circles indicate points pertaining to the plane .(δ; ∈) whose corresponding Floquet multipliers are depicted in Fig. 2.9b. For .0 ≤ δ ≤ 2.18 unbounded solutions are observed. This can be also inferred from Fig. 2.9b, which shows that the points .(δ; ∈) = (1.0; 1.5) and .(2.1; 1.5) are associated with at least one Floquet multiplier (.ρ1 or .ρ2 ) outside5 the unit circle. 5

For .(δ; ∈) = (1.0; 1.5), .ρ1 = −7.06, outside the region shown in Fig. 2.9b.

46

G. R. Franzini and G. J. Vernizzi

Notice that, since the point .(δ; ∈) = (2.1; 1.5) is close to the transition curves, the corresponding Floquet multipliers are close to the unit circle through.−1 and indicate that this transition curve is associated with a period-doubling bifurcation. The point .(δ; ∈) = (3.7; 1.5) is located inside a region of bounded responses, but close to the transition curve (see Fig. 2.9a). Considering the representation of the Floquet multipliers in the Argand’s plane shown in Fig. 2.9b, this point has .ρ1 and .ρ2 on the unit circle, representing stability of the periodic orbit. When .δ is increased to .4.0, the Floquet multipliers leave the unit circle through .+1. Figure 2.9c brings the evolution of the Floquet multipliers along the line .∈ = 1.5 shown in Fig. 2.9a. In Fig. 2.9c, the real and imaginary part of the Floquet multipliers are plotted and the colorscale represents the value of .δ. If a color appears outside the unit circle, the corresponding value of .δ is associated with unbounded responses. On the other hand, stability is observed for the values of .δ associated with colors appearing on the unit circumference. If damping is considered in the mathematical model, points associated with bounded responses are characterized by .|ρ1,2 | < 1, indicative of asymptotic stability of the periodic orbits.

2.6 Examples of Application on the Dynamics of Slender Structures In terms of engineering applications, marine structures are a plentiful field for the occurrence of parametric excitation. The floating units such as oil and gas exploration platforms or transport ships are constantly under the action of sea waves, which generate motion of the floating unit on the vertical plane. This kind of response cause imposed top-motions on slender structures such as risers, mooring lines and umbilical cables, depending on the application. The component of the top motion aligned with the axial direction in such structures generates a temporal variation of the geometric stiffness (i.e., that associated with the tension). Depending on the frequency components and amplitudes of the top motion, scenarios of parametric resonance may appear during the operational lifetime of these structures, which may impact on its durability, with fatigue of the material being an issue.

2.6.1 A Brief Overview of the Literature Different approaches may be taken to investigate the phenomenon, which may present a great variety of behaviors depending on the conditions under analysis. Considering the focus given to systems represented by means of ordinary differential equations in the previous sections, the approach herein detailed is the use of reduced-order models (ROMs) to represent the dynamics of the structure. This approach is well established in the literature, with one of the pioneer studies being [10], where the dynamical

2 An Introduction to Parametrically Excited Systems and …

47

response of a string hanging in fluid under top-motion excitation is investigated. The model considered is linear, asides from the quadratic hydrodynamic damping, that is shown to be essential to limit the amplitudes of lateral vibrations of the string under the parametric instability condition. Following, some different analysis were carried out for tethers, risers and similar slender and flexible structures. In [19] the Mathieu’s instability for a tether under an imposed vertical top-motion is investigated, considering that the tension variation along the structure is negligible. The main focus of the work is to expand the known results of stability charts to higher values of the parameters which may occur in marine structures under low tension. The variation of tension along the length is taken into account in [22], and it is shown to have an important effect on the conditions for parametric instability and the resulting dynamic behavior and steady-state amplitude. These results show the importance of the proper consideration of effects that may alter the stiffness of the structure, in this case, the tension along its length. Complementing the studies on slender marine structures, the effects of the axial dynamics are included and investigated in [2, 3]. These studies show the role of multi-modal and internal resonance effects between axial and transversal modes. In particular, it is shown in [2] that the achievement of a steady-state depends on the nonlinear damping. In the presence of only structural nonlinearities, the steady-state is not achieved for the investigated slender structures. An analytical solution for the main parametric resonance is presented in [4] for the problem with interactions from the axial dynamics. Other effects may also be included in the modeling, such as the contribution of a horizontal top-motion investigated in [27]. It is shown that the horizontal imposed motion also contributes with parametric terms in the model asides the expected forcing terms. In [26], the effects of irregular waves are considered, showing that they significantly change the stability charts of the structure. Following, the combination of multi-frequency parametric excitation and vortex-induced vibrations (VIV) is investigated in [25]. It is shown that VIV dominates the response in mild sea conditions, while the parametric excitation dominates it in more severe conditions with VIV amplifying the vibration amplitude. Experimental investigations on the parametric excitation of a submerged cylindrical model and on the concomitant effect of parametric excitation and VIV on slender structures can be found in [5, 8].

2.6.2 Reduced-Order Modeling and Application of the Presented Techniques ROMs are particularly interesting to investigate the effects of parametric excitation on slender structures because they are a suitable alternative in place of tackling on the partial differential equations of motion in the continuous domain. A full solution with finite differences or the finite element method (FEM) would require a bigger computational effort and time consumption in relation to a small system of ordinary

48

G. R. Franzini and G. J. Vernizzi

Fig. 2.10 Sketch of the horizontal slender structure problem

differential equations. Another usefulness relies on the smaller amount of parameters to be taken into account in the simulations. One such example is the initial conditions, which consist of a small number of parameters to input, while in a FEM solution these parameters can be varied for each node, easily adding up to a hundred input values on simple problems. Of course there is the drawback that the phenomena that can be catch by the ROM are limited by the modeling technique and limitations employed. Examples of sensitivity analysis of some parameters by the means of ROMs can be found, for example, in [6, 9]. A very common procedure to obtain ROMs is the use of Galerkin projections with a previously chosen set of functions, usually the vibrations modes of the structure. The mathematical steps for the obtaining of ROMs for slender structures is presented in this section, being also a topic well developed in the literature, as can be found in [7, 11, 12, 23]. For the sake of simplicity, a horizontal slender cylinder immersed in still fluid is considered with a time-varying end tension applied at one of the supports, as depicted in Fig. 2.10. For a discussion regarding the reduced order modeling of a cylinder with varying tension along its length, the reader is referred to [23]. The reference frame .x, z is shown with the corresponding displacements on each direction.u, w. The time-varying end-tension is given as.T = T + ΔT cos Ωt, where . T is a pretension applied to the structure and .ΔT is an end-tension variation caused by an imposed displacement .u L with the relationship .ΔT = E Au L /L. In turn, .μ, . E A, . E I and . L are the mass per unit length, axial stiffness, bending stiffness and length of the cylinder respectively. For the problem at hand, it is also necessary to define the specific mass of the fluid .ρ, the cylinder diameter . D, the potential added mass per unit length .μa and the mean drag coefficient .C D . Disregarding the axial dynamics the transversal motion is ruled by Eq. 2.69, while the axial displacements may be then obtained by means of Eq. 2.70 once .w is known. . (μ+μa )

| | ) ( ( ∂4w ∂2w E A ∂ 2 w L ∂w 2 ρDC D || ∂w || ∂w ∂2w = 0, + E I − T − dx + 2L ∂x 2 0 ∂x 2 | ∂t | ∂t ∂t 2 ∂x 4 ∂x 2

(2.69)

u (x, t) =

.

Tx x + E A 2L

( L( 0

∂w ∂x

)2 dx −

1 2

( x( 0

∂w ∂ξ

)2 dξ.

(2.70)

The procedure to obtain these equations using Hamilton’s principle is described in [7, 13, 23], for example. For the set of equations before the disregarding of the axial

2 An Introduction to Parametrically Excited Systems and …

49

dynamics, the reader is referred to [13, 24]. For the case at hand, since the tension T is constant along the beam, varying only with time, the shape of the vibration modes are given by sinusoidal functions. That said, a Galerkin procedure with a single projection function is used, assuming .w(x, t) = ψ(x)Dq(t), with .ψ(x) = sin(πx/L). The inclusion of the diameter as scaling in the trial function is made in order to have .q(t) in dimensionless amplitudes. With some algebraic work, the equation of the ROM is given by:

.

| | ( ( ) ) π 2 d2 q E AD 2 ( π )4 3 4ρD 2 C D dq || dq || (π/L)2 = 0. EI + +T q + q + . μ + μa L 4 (μ + μa ) L 3π (μ + μa ) dt | dt | dt 2

(2.71)

From Eq. 2.71 is possible to obtain the first natural frequency of the structure, namely: ( ( ) ) π 2 (π/L)2 2 EI (2.72) .ω = +T . μ + μa L Using a dimensionless time .τ = Ωt/2, the final form for the equation of motion describing the ROM is: q¨ + (δ + 2∈ cos (2τ )) q + aq 3 + b |q| ˙ q˙ = 0.

.

(2.73)

The correspondence between the parameters present in Eq. 2.73 and the continuous model parameters are then given in Table 2.1. With the obtained form of Eq. 2.73, it is possible to apply any of the techniques presented so far in the analysis of the problem of the slender structure concerning the parametric excitation of the chosen projected mode in the Galerkin procedure. For structures with varying tension along the length, it is also possible to achieve this same format for the ROM, with the drawback of the expressions for the parameters of the final form not being simple due to the form of the vibration modes of the structure. Such a development can be found in [23]. To exemplify the practical applicability of the development made so far, some example cases are taken from the literature for exploration. The first two cases refer Table 2.1 Parameters for the modified Mathieu equation as function of the model parameters

Parameter .δ .∈ .a .b

Evaluation ( ) ω 2 . Ω/2 2ΔT (π/L)2 . (μ + μa ) Ω 2 ( π )4 E AD 2 . Ω 2 (μ + μa ) L 4ρD 2 C D . 3π (μ + μa )

50

G. R. Franzini and G. J. Vernizzi

Table 2.2 Parameters for the structures of Case I [26], Case II [20] and Case III [7]. All values are presented in the international system Case I Case II Case III Parameter .μ .μa .E I .E A .T .L .D .C D

× 102 2 .3.511 × 10 8 .5.458 × 10 10 .1.084 × 10 7 .6.950 × 10 2 .8.610 × 10 −1 .6.604 × 10 0 .1.200 × 10 .7.085

× 102 2 .5.308 × 10 9 .1.457 × 10 10 .1.880 × 10 7 .1.300 × 10 2 .3.000 × 10 0 .0.812 × 10 0 .0.800 × 10 .5.123

× 100 0 .0.397 × 10 0 .0.056 × 10 3 .1.200 × 10 1 .4.000 × 10 0 .2.552 × 10 −1 .0.222 × 10 0 .1.200 × 10 .1.190

to the investigation to whether or not the trivial solution is asymptotically stable. In the case in which it is, a decay with time of the vibration amplitude is expected. However when its not, it is expected that a steady-state regime is achieved, possible due to the presence of the nonlinear damping. The third case under analysis however, consists of the prediction of the post-critical response amplitudes as a function of the tension variation with time and the frequency of that excitation. Naming these cases as “Case I”, “Case II” and “Case III” respectively, the relevant properties of the structures can be found in Table 2.2. Since the common case in the literature for submerged structures is the presence of tension variation along the length and we are interested in constant values, the top-tension of each case was assumed as this constant value to be used in the analysis carried out. For all cases, the sea-water specific mass is taken as .ρ = 1025 kg/m.3 . For Cases I and II, let consider the relationship between the parametric excitation frequency and the structure’s natural frequency to be .Ω = 1.9ω. Using the expressions in Table 2.1, the mentioned condition leads to .δ = 1.108. Using then Eq. 2.42 up to order .∈ for simplicity, we get .∈ = 0.108 for the transition of the stability of the trivial solution. Considering then values of .∈ .5% above and below the limit case, that is,.∈ = 0.113 and.∈ = 0.103, it is expected that in one case a steady-state solution will appear, while in the other the vibrations decay with time. For Case I, .∈ = 0.103 leads to .ΔT = 12877 kN while .∈ = 0.113 leads to .ΔT = 14232 kN. For the case with vibrations predicted to decay with time a time-series of the amplitude with respect to the dimensionless time is shown in Fig. 2.11, comparing the integration of the ROM with a solution using FEM. The time-series of the middle point of the structure is considered in the case of FEM simulations, since we are focusing on vibrations of the first mode, which presents its maximum in the middle length for structures under a constant tension along the length. The software Giraffe has been used to perform the FEM simulations. Giraffe is an in-house software for finite element simulations developed by Prof. Alfredo Gay Neto at Escola Politécnica, University of São Paulo, being already used in previous research efforts associated with offshore structures. Details regarding Giraffe and the references for its elements formulations can be

2 An Introduction to Parametrically Excited Systems and …

51

0.4 ROM FEM

Dimensionless Amplitude

0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4

0

50

100

150

200

250

300

350

400

450

500

550

Fig. 2.11 Comparison between ROM and FEM simulations for a set .(δ, ∈) = (1.108, 0.103) predicted to lead to zero amplitude by the analytical solutions 0.4 ROM FEM

Dimensionless Amplitude

0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4

0

50

100

150

200

250

300

350

400

450

500

550

Fig. 2.12 Comparison between ROM and FEM simulations for a set .(δ, ∈) = (1.108, 0.113) predicted to lead to finite post-critical amplitude by the analytical solutions

found in [18]. In turn, the ROM is integrated using the native ode45 function from Matlab® . It can be seen in Fig. 2.11 that both the ROM and FEM solutions decay with time, as expected from the analytical predictions. Following, the simulations with .∈ = 0.113 for Case I are shown in Figs. 2.12 (full simulation) and 2.13 (focused on the steady-state regime). As it is shown, both solutions present the same frequency component on the time-series and both developed a steady-state regime. There is a small difference in the amplitudes which is not surprisingly considering the comparison is being made between a solution with a single modal projection function against a richly discretized continuous model. The match is expected to become better as richer

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Dimensionless Amplitude

ROM FEM

0.2 0.1 0 -0.1 -0.2 -0.3 400

450

500

550

Fig. 2.13 Comparison between ROM and FEM simulations for a set .(δ, ∈) = (1.108, 0.113) predicted to lead to finite post-critical amplitude by the analytical solutions. Focus on the Steady-state regime 0.3

Dimensionless Amplitude

ROM FEM

0.2 0.1 0 -0.1 -0.2 -0.3

0

50

100

150

200

250

300

350

400

450

500

550

Fig. 2.14 Comparison between ROM and FEM simulations for a set .(δ, ∈) = (1.108, 0.103) predicted to lead to zero amplitude by the analytical solutions

ROMs are used, which may include the axial dynamics as well. Notice however, that all the qualitative aspects and an already good amplitude prediction are obtained with much less computational effort. The FEM solution required around 10 min to complete against mere 2 seconds for the ROM integration in the same computer (i7 7th generation, 16Gb RAM). Finally, the computations needed to predict the qualitative behavior with the analytical computations can be easily done in a standard scientific calculator or a spreadsheet software. Now, for Case II, using the expressions in Table 2.1, the tension variation for .∈ = 0.103 leads to .ΔT = 2438 kN and .∈ = 0.113 leads to .ΔT = 2695 kN. The timeseries for the scenario with expected decaying vibrations are shown in Fig. 2.14.

2 An Introduction to Parametrically Excited Systems and …

53

0.4 ROM FEM

Dimensionless Amplitude

0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4

0

50

100

150

200

250

300

350

400

450

500

550

Fig. 2.15 Comparison between ROM and FEM simulations for a set .(δ, ∈) = (1.108, 0.113) predicted to lead to finite post-critical amplitude by the analytical solutions 0.2

ROM FEM

Dimensionless Amplitude

0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 400

450

500

550

Fig. 2.16 Comparison between ROM and FEM simulations for a set .(δ, ∈) = (1.108, 0.113) predicted to lead to finite post-critical amplitude by the analytical solutions. Focus on the Steady-state regime

Again, the obtained results agree with the behavior expected from the analytical prediction. Both the ROM and the FEM simulations present an amplitude of vibration decaying with time. In the sequence, the simulations with the set expected to produce a steady-state regime are shown in Figs. 2.15 (full simulation) and 2.16 (focused on the steady-state regime). Again, an excellent match in the qualitative behavior is achieved, with the same frequency components and steady-state achieved. There is also a very good quantitative match in the vibration amplitude between ROM and FEM solutions. Now, for Case III, the development made in Sect. 2.3.2 is used. Three different scenarios for the variation of tension are considered, namely, .ΔT equal to .20, .30 and .40% of the average tension .T . In terms of practical applications, the .40% condition is already

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Dimensionless Amplitude

1.5 20% Analytical 20% ROM 20% FEM 30% Analytical 30% ROM 30% FEM 40% Analytical 40% ROM 40% FEM

1

0.5

0

0

0.5

1

1.5

2

2.5

3

Fig. 2.17 Comparison between ROM, FEM and analytical solutions for the steady-state amplitude as a function of the parametric excitation frequency for three amplitudes of tension variation, namely .20, .30 and .40% of the average tension

a large value. The steady-state amplitudes of vibration are then obtained for each case for a series of values of the relation .Ω/ω, comparing the use of Eq. 2.54, the numerical integration of the ROM and the FEM solution. These results are shown in Fig. 2.17. As it can be seen, the match between the analytical solution and the integration of the ROM is almost perfect. In turn, there is a quantitative difference between the FEM simulations and the analytical or ROM solutions. The relative difference stays around .10% on the peak values, which keeps the analytical solution as a suitable evaluation for the post-critical amplitude. It is important to highlight that each point of evaluation of the FEM solution takes about 10 min in a standard office computer (i7 7th generation, 16Gb RAM), while the full map with the analytical solution takes only a fraction of second to be evaluated. This shows how the analytical investigation of the modified Mathieu’s equation with nonlinear terms can be used in the practice of engineering design of slender structures. The analytical solution recovers the qualitative aspects of the problem, as can be seen by the shapes of the curves in Fig. 2.17 in relation to the FEM obtained points, while also bringing a good estimate for the expected vibration amplitudes in the post-critical condition. These results reinforce how the analytical solution can be used as a valuable tool, specially in early stages of structural design, helping to delimit the range of parameters for further investigations using more computationally costly tools such as FEM simulations.

2 An Introduction to Parametrically Excited Systems and …

55

Acknowledgements The first author is grateful to Brazilian Research Council (CNPq) for the grant 305945/2020-3. The second author thanks São Paulo State Research Agency (FAPESP) for his Ph.D. scholarship (grants 2016/25457-1 and 2017/16578-2).

References 1. Bender, C.M., Orszag, S.A.: Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill (1978) 2. Chatjigeorgiou, I.: On the parametric excitation of vertical elastic slender structures and the effect of damping in marine applications. Applied Ocean Research 26(1-2), 23–33 (2004). https://doi.org/10.1016/j.apor.2004.08.001 3. Chatjigeorgiou, I.K., Mavrakos, S.A.: Bounded and unbounded coupled transverse response of parametric excited vertical marine risers and tensioned cable legs for marine applications. Applied Ocean Research 24, 341–354 (2002) 4. Chatjigeorgiou, I.K., Mavrakos, S.A.: Nonlinear resonances of parametrically excited risers numerical and analytic investigation for .Ω = 2ω1 . Computers & Structures 83(8), 560 – 573 (2005). https://doi.org/10.1016/j.compstruc.2004.11.009 5. Franzini, G., Pesce, C., Gonçalves, R., Fujarra, A., Mendes, P.: An experimental investigation on concomitant vortex-induced vibration and axial top-motion excitation with a long flexible cylinder in vertical configuration. Ocean Engineering 156, 596–612 (2018). https://doi.org/10. 1016/j.oceaneng.2018.02.063 6. Franzini, G.R., Dias, T., Mazzilli, C., Pesce, C.P.: Parametric excitation of an offshore riser using reduced-order models based on Bessel-type modes: assessment on hydrodynamic coefficients effects. In: Proceedings of the 6th International Conference on Nonlinear Science and Complexity. INPE Instituto Nacional de Pesquisas Espaciais (2016). https://doi.org/10.20906/ cps/nsc2016-0009 7. Franzini, G.R., Mazzilli, C.E.N.: Non-linear reduced-order model for parametric excitation of vertical and immersed slender rod. International Journal of Non-linear Mechanics 80, 29–39 (2016). https://doi.org/10.1016/j.ijnonlinmec.2015.09.019 8. Franzini, G.R., Pesce, C.P., Salles, R., Gonçalves, R.T., Fujarra, A.L.C., Mendes, P.: Experimental investigation with a vertical and flexible cylinder in water: response to top motion excitation and parametric resonance. Journal of Vibration and Acoustics 137 (3), 031010–1–031010–12 (2015). https://doi.org/10.1115/1.4029265 9. Franzini, G.R., Santos, C.C.P., Mazzilli, C.E.N., Pesce, C.P.: Parametric excitation of an immersed, vertical and slender beam using reduced-order models: influence of hydrodynamic coefficients. Marine Systems & Ocean Technology 11(1-2), 10–18 (2016). https://doi.org/10. 1007/s40868-016-0013-z 10. Hsu, C.S.: The response of a parametrically excited hanging string in fluid. Journal of Sound and Vibration 39(3), 305–316 (1975). https://doi.org/10.1016/s0022-460x(75)80084-8 11. Mazzilli, C.E., Rizza, F., Dias, T.: Heave-imposed motion in vertical risers: A reduced-order modelling based on Bessel-like modes. Procedia IUTAM 19, 136–143 (2016). https://doi.org/ 10.1016/j.piutam.2016.03.018 12. Mazzilli, C.E.N., Dias, T.: Non-linear reduced-order modelling of heave-imposed motion in vertical risers. In: Proceedings of the 15th Pan-American Congress of Applied Mechanics PACAM XV (2015) 13. Mazzilli, C.E.N., Sanches, C.T., Baracho Neto, O.G.P., Wiercigroch, M., Keber, M.: Non-linear modal analysis for beams subjected to axial loads: Analytical and finite-element solutions. International Journal of Non-linear Mechanics 43, 551–561 (2008). https://doi.org/10.1016/j. ijnonlinmec.2008.04.004 14. Meirovitch, L.: Methods of analytical dynamics. Dover Publications (2003)

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15. Nayfeh, A.H.: Perturbation methods. John Wiley & Sons (1973) 16. Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics - Analytical, Computational and Experimental Methods. John Wiley & Sons (1995) 17. Nayfeh, A.H., Mook, D.T.: Nonlinear oscillations. John Wiley & Sons (1979) 18. Neto, A.G.: Giraffe user’s manual - generic interface readily accessible for finite elements (2021). URL http://sites.poli.usp.br/p/alfredo.gay/giraffe/GIRAFFE_Manual.pdf 19. Patel, M., Park, H.: Dynamics of tension leg platform tethers at low tension. Part i - Mathieu stability at large parameters. Marine Structures 4(3), 257–273 (1991). https://doi.org/10.1016/ 0951-8339(91)90004-u 20. Patel, M., Park, H.: Combined axial and lateral responses of tensioned buoyant platform tethers. Engineering Structures 17(10), 687–695 (1995). https://doi.org/10.1016/01410296(95)00118-q 21. Rand, R.H.: Lecture Notes on Nonlinear Vibrations (2003). URL http://urn.kb.se/resolve? urn=urn:nbn:se:kth:diva-9154 22. Simos, A.N., Pesce, C.P.: Mathieu stability in the dynamics of TLP’s tethers considering variable tension along the length. In: Transactions on the Built Environment, vol. 29 (1997) 23. Vernizzi, G.J., Franzini, G.R., Lenci, S.: Reduced-order models for the analysis of a vertical rod under parametric excitation. International Journal of Mechanical Sciences 163, 105122 (2019). https://doi.org/10.1016/j.ijmecsci.2019.105122 24. Vernizzi, G.J., Lenci, S., Franzini, G.R.: A detailed study of the parametric excitation of a vertical heavy rod using the method of multiple scales. Meccanica 55(12), 2423–2437 (2020). https://doi.org/10.1007/s11012-020-01247-6 25. Yang, H., Xiao, F.: Instability analyses of a top-tensioned riser under combined vortex and multi-frequency parametric excitations. Ocean Engineering 81, 12–28 (2014). https://doi.org/ 10.1016/j.oceaneng.2014.02.006 26. Yang, H., Xiao, F., Xu, P.: Parametric instability prediction in a top-tensioned riser in irregular waves. Ocean Engineering 70, 39–50 (2013). https://doi.org/10.1016/j.oceaneng.2013.05.002 27. Zeng, X., Xu, W., Li, X., Wu, Y.: Nonlinear dynamic responses of the tensioned tether under parametric excitations. in: Proceedings of the 18th International Offshore and Polar Engineering Conference, ISOPE2008 (2008)

Chapter 3

Normal Forms Alois Steindl

Abstract Normal Form Theory helps to simplify linear and nonlinear dynamical systems by reducing the number of terms in the equations and by introducing a symmetry in the system. It is demonstrated, how this Normal Form reduction is applied depending on the properties of the Jordan Normal Form for the linearized system.

3.1 Introduction Given a system of differential equations .

x˙ = Jx + f (x),

(3.1)

for . x(t) ∈ Rn with . x˙ = d x/dt, where the Jacobian matrix .J is already in Jordan Normal Form and . f (x) contains the non-linearities, we try to find a near-identity coordinate transformation . x |→ y x = y + h( y),

(3.2)

˙y = J y + ˜f ( y),

(3.3)

.

such that the transformed system .

is simpler than Eq. (3.1). Ideally one would find . ˜f ( y) = 0, such that Eq. (3.3) is linear, or the original Eqs. (3.1) are “linearized”. Inserting . x = y + h( y) into . x˙ = Jx + f (x) and using .˙ y = J y + ˜f ( y) we obtain the equation

A. Steindl (B) Institute for Mechanics and Mechatronics, TU Wien, Getreidemarkt 9, 1060 Vienna, Austria e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. R. Castilho Piqueira et al. (eds.), Lectures on Nonlinear Dynamics, Understanding Complex Systems, https://doi.org/10.1007/978-3-031-45101-0_3

57

58

A. Steindl

.

) ∂( y + h( y)) ( J y + ˜f ( y) = J ( y + h( y)) + f ( y + h( y)). ∂y

(3.4)

At leading order we obtain the homological equation .

∂h( y) J y − Jh( y) + ˜f ( y) = f ( y), ∂y

(3.5)

which is a partial differential equation for the unknown function . h. It will be solved by power series expansion. The homological operator adJ y h =

.

∂h( y) J y − Jh( y), ∂y

(3.6)

is a linear differential operator and agrees with the Lie bracket between the vector fields .J y and . h( y): adg h = [g( y), h( y)] =

.

∂g( y) ∂h( y) g( y) − h( y). ∂y ∂y

(3.7)

3.1.1 Normal Forms in the Case of a Diagonal Jacobian .J We first assume that .J is diagonal J = diag(λ1 , λ2 , . . . , λn ).

(3.8)

.

For . f and . h we use the multi-index notation f ( y) =

∑

. i

f i,m y m ,

h i ( y) =

∑

m

h i,m y m ,

(3.9)

m

with .m = (m 1 , m 2 , . . . , m n ), .0 ≤ m j ∈ N, .|m| =

∑ j

m j , and

y m = y1m 1 y2m 2 · · · ynm n ,

(3.10)

adJ y y m ei = ((λ1 m 1 + · · · + λn m n ) − λi ) y m ei ,

(3.11)

.

then .

and the homological equation can be solved term by term (λ · m − λi )h i,m = f i,m − f˜i,m .

.

(3.12)

3 Normal Forms

59

If .λ · m − λi /= 0, then h

. i,m

=

f i,m , λ · m − λi

else = 0,

h

. i,m

f˜i,m = 0,

f˜i,m = f i,m .

(3.13)

(3.14)

Since both .h i,m and . f i,m are the coefficients of . y m in the .i-th equation, only those monomials in . f i ( y) survive the Normal Form reduction, for which .m · λ = λi . Therefore the function . ˜f ( y) satisfies adJ y ˜f ( y) = 0.

.

(3.15)

Definition 3.1 The .n-tuple .λ = (λ1 , . . . , λn ) of eigenvalues is said to be resonant, if among the eigenvalues there exists an integral relation of the form λ = m · λ,

. i

(3.16)

∑ ∑ where .m j ≥ 0, . m j ≥ 2. .|m| = m j is called the order of the resonance. Example • The relation .λ1 = 2λ2 is a resonance of order 2; • The relation .2λ1 = 3λ2 is not a resonance; • The relation .λ1 + λ2 = 0 is a resonance of order 3: it implies the resonance .λ1 = 2λ1 + λ2 . If among the eigenvalues .λi no resonance exists, all non-linear terms in . f can be eliminated. Theorem 3.1 (Poincaré ([1])) If the eigenvalues of the matrix .J are nonresonant, then the equation ˙ = Jx + f (x) .x (3.17) can be reduced to the linear equation .

x˙ = Jx

(3.18)

by a formal change of variables . x = y + h( y). Normal Form for a Simple Resonant Problem: .1 : 2-Resonance For the second order resonance .λ2 = 2λ1 the single solution of Eq. (3.15) is given by . f 2,20 y12 and the Normal Form equations become y˙ = λ1 y1 ,

(3.19)

. 1

y˙2 = 2λ1 y2 +

f 2,20 y12 ,

(3.20)

60

A. Steindl

with solution y (t) = y1 (0) exp(λ1 t),

(3.21)

. 1

y2 (t) = y2 (0) exp(2λ1 t) +

f 2,20 y12 (0)t

exp(2λ1 t).

(3.22)

Once the linear equation for . y1 has been solved, the quadratic entry . y12 turns into a (resonant) inhomogeneous part in the equation for . y2 . Another Simple Resonant Problem: 1:-2-Resonance The relation .λ2 = −2λ1 corresponds to the fourth order resonances λ = 3λ1 + λ2 , or λ2 = 2λ2 + 2λ1 .

. 1

(3.23)

In the Normal Form only those monomials . y m occur, for which m 1 − 2m 2 = 1, m 1 − 2m 2 = −2,

(equation for y1 ), (equation for y2 ),

.

y˙ = λ1 y1 +

∞ ∑

. 1

f˜1,2k+1,k y12k+1 y2k = y1 g1 (y12 y2 ),

(3.24) (3.25)

(3.26)

k=1

y˙2 = −2λ1 y2 +

∞ ∑

f˜1,2k−2,k y12k−2 y2k = y2 g2 (y12 y2 ).

(3.27)

k=2

Now both equations contain infinitely many non-linear terms, but these have a special structure: they can be written as simple multiples of some functions .gi (u) with .u = y12 y2 . This structure allows us to solve the equations: The function .u = y12 y2 satisfies the ODE u˙ = 2y1 y2 y˙1 + y12 y˙2 = u(2g1 (u) + g2 (u)),

.

{

so .

G(u) =

du = t − t0 , u(2g1 (u) + g2 (u))

(3.28)

(3.29)

and .u(t) = G −1 (t − t0 ). Using (26) and (27) we can now solve for . y1 (t) and . y2 (t): { .

t

y j (t) = y j (t0 ) exp

g j (u(τ ))dτ .

(3.30)

t0

The Normal Form reduction has rendered the differential equations solvable.

3 Normal Forms

61

Fig. 3.1 A vector field . f ( y) is equivariant w.r.t. a linear transformation . y | → v = B y, if .B f ( y) = f (v)

3.1.1.1 Digression: Symmetry of a Differential Equation Definition 3.2 (Equivariance and Invariance) Let .G be a matrix group of transformations . y | → v = B y, with B ∈ G. (3.31) A vector field . f ( y) is equivariant w.r.t. .G, if (Fig. 3.1) .

f (B y) = B f ( y), for all B ∈ G and all y.

(3.32)

A function . f ( y) is invariant, if f (B y) = f ( y).

.

(3.33)

Example: Symmetry for 1:-2 Resonance The set of matrices B(s) =

.

) ( ( s ) 1 0 e 0 = exp Js, with J = , 0 e−2 s 0 −2

(3.34)

forms a group with .B(0) = I2 , .B(s1 )B(s2 ) = B(s1 + s2 ) and .B−1 (s) = B(−s). An invariant function under the transformation . y |→ B(s) y is given by .u = y12 y2 and the equivariant functions are just the Normal Form equations . f ( y) satisfying adJ y f ( y) = 0.

.

(3.35)

62

3.1.1.2

A. Steindl

Oscillations: Systems with Purely Imaginary Eigenvalues

For purely imaginary eigenvalues .λ1,2 = ±iω it is convenient to use complex coordinates .z = y1 + i y2 , .z = y1 − i y2 and set .

z = w + h(w, w).

(3.36)

The matrix .J is still diagonal and the homological equation for the Normal Form in .w reads . ((m 1 − m 2 − 1)iω) h 1,m 1 ,m 2 = f 1,m 1 ,m 2 − f˜1,m 1 ,m 2 , (3.37) with solution . f˜1 (w, w) =

∞ ∑

f˜1,k+1,k w(ww)k .

(3.38)

k=1

All even order monomials can be eliminated and only odd terms of the form w(ww)k occur in the Normal Form.

.

3.1.1.3

Correction of Higher Order Terms

The calculation of the Normal Form is performed iteratively: quadratic terms, cubic terms, ... The elimination of the quadratic terms also contributes higher order terms. These must be corrected before proceeding with the eliminations. Rewrite Eq. (3.4) in the following form J y + ˜f ( y) + h, ( y)J y + h, ( y) ˜f ( y)

.

= J y + Jh( y) + f ( y) + ( f ( y + h( y)) − f ( y)) ,

(3.39)

where . h, ( y) = ∂h( y)/∂ y. • The terms .J y cancel, • the entries . ˜f ( y) + h, ( y)J y = Jh( y) + f ( y) are the homological equation, • the higher order terms are ignored in the first step, but introduce higher order terms, which have to be taken care of in the next steps. At second order we obtain the solutions of the homological Eq. (3.5) for the oscillations

3 Normal Forms

63

h

. 1,2,0

h 1,1,1 h 1,0,2

f 1,2,0 , iω f 1,1,1 = , −iω f 1,0,2 = , −3iω =

f˜1,2,0 = 0,

(3.40)

f˜1,1,1 = 0,

(3.41)

f˜1,0,2 = 0,

(3.42)

and h (z, z) = h 1 (z, z).

. 2

(3.43)

From the quadratic terms f (2) (z, z) = f 1,2,0 z 2 + f 1,1,1 zz + f 1,0,2 z 2 = f 1(2) (w + h 1 (w, w), w + h 2 (w, w)), (3.44)

. 1

we obtain the new cubic terms fˆ(3) = f 1,2,0 (2wh 1 ) + f 1,1,1 (wh 2 + wh 1 ) + f 1,0,2 (2wh 2 ),

. 1

(3.45)

especially the additional coefficient of .w 2 w becomes fˆ

= 2 f 1,2,0 h 1,1,1 + f 1,1,1 (h 1,1,1 + h 1,2,0 ) + 2 f 1,0,2 h 1,0,2 2 1 1 2 = f 1,2,0 f 1,1,1 + f 1,1,1 f 1,1,1 + f 1,1,1 f 1,2,0 + f 1,0,2 f 1,0,2 . −iω iω iω 3iω . 1,2,1

(3.46)

We also obtain new contributions to fourth and higher order terms: • From the quartic terms f

. 1,2,0

h 21 + f 1,1,1 h 1 h 2 + f 1,0,2 h 22

(3.47)

and 3w2 h 1 + f 1,2,1 (w 2 h 2 + 2wwh 2 ) + f 1,1,2 (2wwh 2 + w 2 h 1 ) + f 1,0,3 3w 2 h 2 , (3.48) • and from solving the equation f

. 1,3,0

(1 + h, (w)) ˜f = f (w + h(w)) − f (w).

.

(3.49)

In most cases the cubic terms in the Normal Form are sufficient for investigating the dynamics of the system, and these higher order terms need not be determined. In the case of 2 pairs of purely imaginary eigenvalues.±iω1 ,.±iω2 the homological 2 m3 m4 equations for the monomials .z 1m 1 z m 1 z 2 z 2 are

64

A. Steindl .

(m 1 − m 2 )iω1 + (m 3 − m 4 )iω2 = iω1 , for 1st equation,

(3.50)

(m 1 − m 2 )iω1 + (m 3 − m 4 )iω2 = iω2 , for 2nd equation.

(3.51)

In the non-resonant case.ω1 /ω2 ∈ / Q Eqs. (3.50) and (3.51) admit only monomials, for which .

m 1 = m 2 + 1,

m 3 = m 4 for Eq. (50),

(3.52)

m 3 = m 4 + 1,

m 1 = m 2 for Eq. (51).

(3.53)

At third order we find the leading terms z˙ = iω1 z 1 + (A1 |z 1 |2 + A2 |z 2 |2 )z 1 ,

(3.54)

z˙ 2 = iω2 z 2 + (A4 |z 1 | + A5 |z 2 | )z 2 ,

(3.55)

. 1

2

2

with . A j = c j + id j ∈ C, e.g. . A1 = f˜1,2,1,0,0 . Only 2 out of originally 20 cubics per equation survive the Normal Form reduction. Introducing polar coordinates .z j = r j exp(iϕ j ), the system of Eqs. (3.54) and (3.55) becomes r˙ = (c1r12 + c2 r22 )r1 ,

(3.56)

. 1

r˙2 =

(c4 r12

ϕ˙ 1 = ω1 + ϕ˙ 2 = ω2 +

+ c5r22 )r2 , d1r12 + d2 r22 , d4 r12 + d5r22 .

(3.57) (3.58) (3.59)

We observe that the angular variables do not show up on the right hand side at all and it is sufficient to investigate only the system for the radii .r j . In the case of resonances the reduced system will not be that simple: if the two frequencies .ω1 and .ω2 satisfy the relation ω1 : ω2 = k : l, with k, l ∈ Z,

(3.60)

.

we may set .ω1 = kω, .ω2 = lω and the homological equations { k for eqn. 1 .(m 1 − m 2 )k + (m 3 − m 4 )l = , l for eqn. 2

(3.61)

admit the additional solutions (Fig. 3.2) m 1 = m 2 + 1 + l, m 3 = m 4 + 1 + k,

.

m 3 = m 4 + k, m 1 = m 2 + l,

for eqn. 1, for eqn. 2.

(3.62) (3.63)

3 Normal Forms

65

Fig. 3.2 Resonant terms for the .1 : 2-resonance. The leading terms in the Normal Form are 2 .z˙ 1 = iωz 1 + (A1 |z 1 | + A2 |z 2 |2 )z 1 + A3 z 1 z 2 , and 2 .z˙ 2 = 2iωz 2 + (A4 |z 1 | + A5 |z 2 |2 )z 2 + A6 z 12

Observations for 2-DOF Oscillators • The entries iω1 z 1 + (A1 |z 1 |2 + A2 |z 2 |2 )z 1 ,

(3.64)

iω2 z 2 + (A3 |z 1 | + A4 |z 2 | )z 2 ,

(3.65)

.

2

2

occur in all cases, • for the .1 : 2-resonance the nonlinearities start at second order, • for the .1 : 3 and .1 : 1-resonance the lowest resonant terms start at third order, • for other resonances the resonant terms start at order > 3. In the case of . N pairs of purely imaginary eigenvalues .±iω j the homological equations become N ∑ . i(m 2k−1 − m 2k )ωk = iω j . (3.66) k=1

The frequencies are resonant, if the equation .

m · ω = ωj

(3.67)

is satisfied for some nontrivial tuple .m ∈ Z N . The non-resonant Normal Form equations at leading order read z˙ = (iω j +

N ∑

. j

A j,k |z|2k )z j .

(3.68)

k=1

Remarks About Resonances for Oscillators • For . N > 2 resonances can occur, even if the frequencies .ω j are pairwise incom/ Q. As an example the frequencies mensurable: .ω j /ωk ∈ ω = (1, φ, φ ),

.

2

√ 1+ 5 with φ = (Golden Ratio) 2

(3.69)

66

A. Steindl

are incommensurable, but satisfy the relation φ2 − φ − 1 = ω3 − ω2 − ω1 = 0.

(3.70)

.

• For “generic” systems no resonances occur. • Symmetries may enforce multiple eigenvalues.

3.1.2 Non-semisimple Jordan Normal Forms If the Jordan Normal Form matrix .J is non-diagonal, also the homological equation becomes non-diagonal. As simple example we consider the Takens-Bogdanov bifurcation .

x˙ = Jx + f (x), with J =

( ) 01 , 00

(3.71)

with cubic nonlinearities. Choosing the basis vectors for . h ( 3) y1 , 0 ( ) 0 , b5 = y13 b =

. 1

) y12 y2 , 0 ) ( 0 b6 = , y12 y2 (

b2 =

) y1 y22 , 0 ) ( 0 b7 = , y1 y22 (

b3 =

( 3) y2 , 0 ( ) 0 b8 = , y23

b4 =

(3.72) (3.73)

and calculating the action of .adJ y on these basisvectors, e.g. adJ y b1 = 3b2 ,

.

adJ y b5 = 3b6 − b1 ,

(3.74)

and comparing the coefficients in the equation, we obtain the linear system ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ .⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ~

0 3 0 0 0 0 0 0

0 0 2 0 0 0 0 0

0 0 0 1 0 0 0 0

⎞⎛ ⎞ ⎛ ⎞ ⎛˜ ⎞ f 1,3,0 f 1,3,0 0 −1 0 0 0 h 1,3,0 ⎜ f˜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ 0 0 −1 0 0 ⎟ ⎜h 1,2,1 ⎟ ⎜ f 1,2,1 ⎟ ⎜ 1,2,1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ f˜1,1,2 ⎟ 0 0 0 −1 0 ⎟ ⎟ ⎟ ⎜h 1,1,2 ⎟ ⎜ f 1,1,2 ⎟ ⎜ ⎜˜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 0 0 0 −1 ⎟ ⎜h 1,0,3 ⎟ ⎜ f 1,0,3 ⎟ f ⎜ ⎟ 1,0,3 ⎟ ⎟ . (3.75) ⎜h 2,3,0 ⎟ = ⎜ f 2,3,0 ⎟ − ⎜ ˜ 0 0 0 0 0 ⎟ ⎜ f ⎟⎜ ⎟ ⎜ ⎟ ⎜ 2,3,0 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ 0 3 0 0 0 ⎟ ⎜h 2,2,1 ⎟ ⎜ f 2,2,1 ⎟ ⎜ f˜2,2,1 ⎟ ⎜ ⎟ 0 0 2 0 0 ⎠ ⎝h 2,1,2 ⎠ ⎝ f 2,1,2 ⎠ ⎝ f˜2,1,2 ⎠ 0 0 0 1 0 h 2,0,3 f 2,0,3 f˜2,0,3 ~~ ~ AJ y

3 Normal Forms

67

Fig. 3.3 The coefficient of . b5 can be used to select a convenient place for . ˜f by shifting . f along the direction of .adJ y b5 .h 2,3,0

The entry .(AJ y )i j is the coefficient of . bi in .adJ y b j . In the matrix .AJ y row.5 is zero and .row6 = −3 row1 . Therefore the entry . f 2,3,0 b5 cannot be modified at all and the coefficients of . b1 and . b5 cannot both be eliminated by a proper choice of .h 2,3,0 , unless . f 2,2,1 = −3 f 1,3,0 . As shown in Fig. 3.3, different convenient choices for the Normal Form are possible by varying the coefficient .h 2,3,0 . A convenient choice for the Normal Form of the Takens-Bogdanov bifurcation is given by requiring that the first equation contains no non-linear terms. From. f˜1,3,0 = 0 we find .h 2,3,0 = − f 1,3,0 and . f˜2,2,1 = f 2,2,1 − 3h 2,3,0 = f 2,2,1 + 3 f 1,3,0 . Since the remaining six terms can be eliminated, this leads to the differential equations y˙ = y2 , y˙2 = f˜2,3,0 y13 + f˜2,2,1 y12 y2 ,

(3.76)

. 1

with f˜2,3,0 = f 2,3,0 ,

(3.77)

or .

y¨ = f˜2,3,0 y 3 + f˜2,2,1 y 2 y˙

with y = y1 .

(3.78)

Alternatively one could set.h 2,3,0 = f 2,2,1 /3 and. f˜1,3,0 = f 1,3,0 + h 2,3,0 and obtain the system (Fig. 3.3) y˙ = y2 + f˜1,3,0 y13 , y˙2 = f˜2,3,0 y13 .

. 1

(3.79) (3.80)

68

A. Steindl

3.1.2.1

The Elphick-Tirapegui-Bachet-Coullet-Iooss Normal Form ([2])

By introducing an appropriate inner product between polynomials ( p(x), q(x)) N := p(∂)q(x)| x=0 ,

.

(3.81)

the calculation of the Normal Form can be simplified and the Normal Form equations ˜f become symmetric. ∑ If . p(x) = m am x m , then

.

.

∑

p(∂)q(x) =

am 1 ,m 2 ,...,m n

m 1 ,m 2 ,...,m n

∂ m1 ∂ m2 ∂ mn q(x). m1 m2 · · · ∂x1 ∂x2 ∂xnm n

(3.82)

For monomials we find { (x , x ) N =

.

m

m

m 1 !m 2 ! · · · m n ! if m = m, 0 else.

(3.83)

The definition can be simply extended to vector valued polynomials: If . p(x) = ( p1 (x), . . . , pn (x)T and .q(x) = (q1 (x), . . . , qn (x)T , then ( p(x), q(x)) N =

n ∑

.

( pi (x), qi (x)) N .

(3.84)

i=1

With this inner product we can define the adjoint matrix (A⋆ x, y) N = (x, A y) N for all x, y.

.

(3.85)

It is related to the transposed matrix .AT by T A∗ = M−1 N A MN ,

.

(3.86)

where .M N is the metric tensor defined by (x, y) N = x T M N y.

.

(3.87)

Theorem 3.2 (Elphick, Tirapegui, Bachet, Coullet, Iooss) The adjoint of .adJ y is given by .adJT y . The orthogonal complement (w.r.t. the introduced norm) of .Im (adJ y ) is given by .Ker adJT y . If the Normal Form . ˜f is chosen in .Ker adJT y , it is equivariant w.r.t. the group generated by the flow of .JT y.

3 Normal Forms

69

Proceeding as before, we calculate the for .JT : ⎛ 0 1 0 0 ⎜ 0 0 2 0 ⎜ ⎜ 0 0 0 3 ⎜ ⎜ 0 0 0 0 .AJT y = ⎜ ⎜−1 0 0 0 ⎜ ⎜ 0 −1 0 0 ⎜ ⎝ 0 0 −1 0 0 0 0 −1

matrix for the holonomical equation ⎞ 0 0 0 0 0 0 0 0 ⎟ ⎟ 0 0 0 0 ⎟ ⎟ 0 0 0 0 ⎟ ⎟, (3.88) 0 1 0 0 ⎟ ⎟ 0 0 2 0 ⎟ ⎟ 0 0 0 3 ⎠ 0 0 0 0

with kernel spanned by .φ1 = e5 and .φ2 = e1 + e6 . One can easily see that the metric tensor in this example is M N = diag(6, 2, 2, 6, 6, 2, 2, 6),

.

(3.89)

because e.g. .∂ y31 y13 = 6 and .∂ y21 ∂ y2 y12 y2 = 2. The symmetry conforming Normal Form at leading order is given by ) Ay13 , with A = f˜1,3,0 = (3 f 1,3,0 + f 2,2,1 )/4, and B = f˜2,3,0 = f 2,3,0 . By13 + Ay12 y2 (3.90) The calculation of the coefficient . f˜1,3,0 = f˜2,2,1 from the original coefficients . f i,m can be either performed by solving the equations .˜ f=

(

− h 2,3,0 = f 1,3,0 − f˜1,3,0 , = f 2,2,1 − f˜2,2,1 = f 2,2,1 − f˜1,3,0 , .

3h 2,3,0

(3.91) (3.92)

which also yields .h 2,3,0 = ( f 2,2,1 − f 1,3,0 )/4, or by projecting . f onto the kernel .φ2 of .AJT y exploiting the inner product: With . ˜f = f˜1,3,0 φ2 we obtain by taking the inner product of the homological equation with .φ2 0 = (φ2 , AJ y ) N = (φ2 , f − ˜f ) N = (φ2 , f ) N − f˜1,3,0 (φ2 , φ2 ) N ,

.

(3.93)

giving .6 f 1,3,0 + 2 f 2,2,1 − 8 f˜1,3,0 = 0. The Normal Form for the Takens-Bogdanov bifurcation is equivariant w.r.t. to flow generated by .JT : ( ) 10 B(s) ˜f ( y) = ˜f (B(s) y), with B(s) = exp sJT = . s1

.

(3.94)

Unfortunately the symmetry doesn’t help to solve the differential equations in this example, because the linear part .J y isn’t equivariant:

70

A. Steindl

B(s)J y =

.

) ( )( )( ) ( ) ( y2 sy1 + y2 10 01 y1 = JB(s) y. = /= y2 sy2 0 s1 00

(3.95)

This inequality can also be shown using Lie brackets (

) 1 0 .[J y, J y] = JJ y − J J y = y /= 0. 0 −1 T

T

T

(3.96)

For this example the Normal Form helps to reduce the number of nonlinear terms, but doesn’t introduce a “useful” symmetry. Summary: Normal Forms for Non-linear Terms • For diagonal Jordan matrices .J the homological equation is diagonal. The Normal Form equations introduce a symmetry into the equations. • For non-diagonal Jordan matrices the homological equations are also non-diagonal. One has to choose a proper basis in the complement of the image space of the operator .adJ y . The method by Elphick et al. [2] can be applied, to simplify the determination of the Normal Form equations and choose a basis to make the Normal Form equivariant w.r.t. the flow generated by .JT . • The homological equations are solved order by order. After determining the Normal Form at order .k one has to calculate the influence of the coordinate transform on the higher order terms before proceeding to order .k + 1.

3.1.3 Normal Form for matrices Varying parameters in the system can introduce perturbations into the Jordan Normal form. At leading order in the perturbation parameter .ε the linear system takes the form ˙ = (J0 + εJ1 )x, .x (3.97) where .J0 is the unperturbed matrix in Jordan Normal Form, .ε 1/4 (left), resp. . K = 1/4 (right). At . K = 1/4 the synchronous solution . I1 = I2 = 1/2, .ϕ1 − ϕ2 = π loses stability Fig. 3.6 Contours of the reduced Hamiltonian .G in Eq. (3.213) for . I1 + I2 = 1 and . K = 1/5 < 1/4. The synchronous solution is unstable and a new pair of stationary rotations is visible

In these coordinates .ψ2 doesn’t appear in the equations and . J2 = H00 = I1 + I2 is a first integral. We can therefore study the dynamics of the Normal Form equations, by looking at the contour lines of .G in the .(I1 , ψ1 )-plane, setting . I1 + I2 = E. In Figs. 3.5 and 3.6 the phase portraits of the reduced system for varying parameter values . K are displayed: At . K = 1/4 a pair of stationary points in the reduced phase space, corresponding to periodic solutions in the original system, bifurcates from the skewsymmetric orbit . I1 = I2 , .ϕ1 − ϕ2 = π.

88

A. Steindl

3.3 Nonlinear Normal Modes In non-linear vibratory systems usually many different persistent motions are observed. The Nonlinear Normal Modes are a special class of persistent solutions, which make it possible to get a clear picture of the corresponding dynamics and to locate these solutions easily. For linear systems ˙ = Ax, .x (3.216) the solutions of the eigenvalue problem Av i = λi v i

(3.217)

.

govern the behaviour of the system. The functions x (t) = xi0 exp(λi t)v i

(3.218)

. i

are the Linear Normal Modes (LNMs). These have some quite simple properties: • The shape of the different LNMs is determined by the corresponding eigenmodes (resp. eigenfunctions) .v i . • Only one linear mode is active for each LNM. • The frequencies do not depend on the amplitude. • The general solution of the system is obtained as superposition of the LNMs. For non-linear systems these properties are usually not fulfilled, but for certain equations special solution configurations are possible: For the classical Nonlinear Normal Modes (NLMs) ([6, 9]) all components perform a synchronous (“unison”) motion: x (t) = ci x1 (t), for i = 2, . . . , n.

. i

(3.219)

As a simple example we reconsider the non-linear conservative 2-dof oscillator in Eqs. (3.206) and (3.207), x¨ + x1 + x13 + K (x1 − x2 )3 = 0,

(3.220)

x¨2 + x2 +

(3.221)

. 1

x23

+ K (x2 − x1 ) = 0. 3

Setting .x2 = cx1 yields x¨ + x1 + x13 + K (1 − c)3 x13 = 0,

. 1

c x¨1 + cx1 +

c3 x13

+ K (c −

1)3 x13

= 0,

comparing the coefficients we obtain the relation (Fig. 3.7)

(3.222) (3.223)

3 Normal Forms 2

c=-1, K>1/4 c=-1, K 1.

(3.228) (3.229)

The variables .(xi , x˙i ) for .i > 1 are expressed as functions . X i (x1 , x˙1 ), . X˙ i (x1 , x˙1 ) of the master coordinates using invariant manifold calculations up to some (low) order. First we rewrite Eq. (3.229) as first order system x˙ = yi ,

(3.230)

. i

y˙i =

−ωi2 xi

− δi yi + f i (x, y),

(3.231)

and set x = X i (x1 , y1 ),

(3.232)

. i

yi = Yi (x1 , y1 ),

for i ≥ 2.

(3.233)

Inserting Eqs. (3.232) and (3.233) into (3.230) and (3.231) yields the equations .

∂ Xi ∂ Xi y1 + (−ω12 x1 − δ1 y1 + f 1 (x1 , X, y1 , Y ) = Yi , ∂x1 ∂ y1 (3.234)

∂Yi ∂Yi y1 + (−ω12 x1 −δ1 y1 + f 1 (x1 ,X,y1 ,Y )=−ωi2 X 1 −δi Yi + f i (x1 ,X,y1 ,Y ). ∂x1 ∂ y1 (3.235) If the nonlinearities . f i start at second order, the initial approximations for . X i and Y also start at this order:

. i

.

X i = ai,1 x12 + ai,2 x1 y1 + ai,3 y12 ,

Yi = ai,4 x12 + ai,5 x1 y1 + ai,6 y12 .

(3.236)

Expressing the leading expansion of . f i in .x1 and . y1 in the same manner f (x1 , 0, y1 , 0) = f i,1 x12 + f i,2 x1 y1 + f i,3 y12 ,

. i

(3.237)

inserting into Eqs. (3.234) and (3.235), and comparing coefficients results in the linear equations for the unknowns .ai,k : ⎛

0 −ω12 0 ⎜ 2 −δ1 −2ω 2 1 ⎜ ⎜ 0 1 −2δ1 .Aa i = ⎜ 2 ⎜ω 0 0 ⎜ i ⎝ 0 ωi2 0 0 0 ωi2

⎞⎛ ⎞ ⎛ ⎞ −1 0 0 ai,1 0 ⎟ ⎜ai,2 ⎟ ⎜ 0 ⎟ 0 −1 0 ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 0 −1 ⎟ ⎟ ⎜ai,3 ⎟ = ⎜ 0 ⎟ . 2 ⎟ ⎜ ⎟ ⎜ ⎟ δi −ω1 0 ⎟ ⎜ai,4 ⎟ ⎜ f i,1 ⎟ 2 ⎠⎝ 2 −δ1 + δi −2ω1 ai,5 ⎠ ⎝ f i,2 ⎠ 0 1 −2δ1 + δi ai,6 f i,3

(3.238)

3 Normal Forms

91

For higher order approximations the necessary calculations become quite involved. In the case of internal resonances in the master coordinates a corresponding multimode invariant manifold is considered.

3.3.2 Nonlinear Normal Modes for Conservative Systems For conservative systems of the form x¨ +

. i

ω2 x 2 ∂V (x) with Potential V = i i + N (x), ∂xi 2

the total energy

1 ∑ 2 x˙ + V (x) = h 2 i=1 i

(3.239)

n

.

E = K +V =

(3.240)

remains constant. By Hamilton’s principle the true evolution . x of a system between two specified states . x(t0 ) = x 0 and . x(t1 ) = x 1 is a stationary point of the action functional { .

S(x) =

t1

L(x(t), x˙ (t), t)dt with the Lagrangian L = K − V.

(3.241)

t0

By an alternative formulation of the variational principle it can be shown that the trajectories between two points . P 1 and . P 2 in phase space are stationary points of the action functional { .

S=

P2

(h − V )1/2 ds with ds 2 =

P1

n ∑

d xi2 ,

(3.242)

i=1

where . S is the length of the trajectory in the Riemannian metric defined by d S 2 = 2(h − V )

n ∑

.

d xi2 .

(3.243)

i=1

The value of the action integral doesn’t depend on the chosen parametrization of the trajectory, therefore we can also choose one of the dependent variables, say . x = x 1 , as parametrization variable. The equations of motion are then given by x ,,

. i

2(h − V (x)) ∂V (x) ∂V (x) ∑n − xi, , i = 2, . . . , n, =− , 2 1 + m=2 (xm ) ∂x ∂xi

(3.244)

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A. Steindl

where .xi = xi (x), i = 2, . . . , n and .xi, = ∂xi /∂x. When the trajectory reaches the Maximum equipotential surface .V (x) = h, the equations become singular and the transversality conditions have to be added x,

. i

∂V (x) ∂V (x) , at x = X, xm = xm (X ), where V (x) = h. =− ∂x ∂xi

(3.245)

At these points the accelerations are obtained as x ,, =

∂ 2 V (x) ∂xi ∂x

+

∑n

, , ∂ V (x) m=2 x m ∂xi ∂xm + x i 2

. i

(

∂ 2 V (x) ∂x 2

+

∑n

, ∂ V (x) m=2 x m ∂x∂xm 2

3 ∂V∂x(x)

) .

(3.246)

The system order decreases by 1, but the obtained differential equations are usually hard to solve analytically. By a scaling .xi |→ εxi the potential .V (x) becomes ε2 . V (x) = 2

( ω1 x 2 +

n ∑

) ωm xm2

+

m=2

∞ ∑

εk N (k) (x),

(3.247)

k=3

where . N (k) contains the contributions of order .k. The equations of motion are x ,,

. i

∂V (x) 2(h − V (x)) ∂V (x) ∑ =− − xi, , 1 + nm=2 xm2 ∂x ∂xi

(3.248)

together with the transversality condition at the maximum equipotential surface ( .

−xi,

∂V (x) ∂V (x) + ∂x ∂xi

) = 0.

(3.249)

x=(X,0)T

The solutions are sought as power series in the parameter .ε: x = xˆi (x) =

∞ ∑

. i

εm xˆi(m) (x),

xˆ1 (x) = x.

(3.250)

m=1

Keeping only the leading term in .ε one obtains the equations xˆ (1) ,, (h 0 − ω12 x 2 /2) − ω12 x xˆi(1) , + ωi2 xˆi(1) +

. i

∂ N (3) (x, 0) = 0, ∂xi

(3.251)

where.h 0 = ω12 X 2 /2 and.∂ N (3) /∂xi (x, 0) = αi x 2 . This linear, non-autonomous, singular differential equation can in principle be solved by hypergeometric functions, but usually a power series solution in .x is preferred.

3 Normal Forms

93

Fig. 3.8 Similar (left) and nonsimilar (right) Nonlinear Normal Modes

Fig. 3.9 NNMs for a symmetric (left) and asymmetric (right) potential

3.3.2.1

Different Types of NNMs

Depending on the potential’s symmetry properties similar and nonsimilar NNMs can occur: The similar NNMs correspond to the “unison” solutions, whereas the relations between the components of the nonsimilar NNMs is non-linear, as displayed in Fig. 3.8. If the potential has a mirror reflection symmetry, the NNMs will be symmetric, otherwise asymmetric. (Fig. 3.9).

3.3.3 Analysis of NNMs Using Group Theory ([3]) The existence of NNMs is related to the equivariance of the equations with respect to continuous and discrete group transformations. As a simple example we consider the case of 2 identical coupled oscillators

94

A. Steindl

x¨ + ω 2 x1 + k(x1 − x2 ) + f NL (x1 − x2 ) = 0,

(3.252)

x¨2 + ω x2 + k(x2 − x1 ) + f NL (x2 − x1 ) = 0,

(3.253)

. 1

2

with . f NL (−x) = − f NL (x), which is equivariant under the reflection .B : x1 ↔ x2 . Introducing symmetry-conforming variables u = (x1 + x2 ), v = (x1 − x2 ), such that Bu = u, Bv = −v,

.

(3.254)

we obtain the decoupled equations u¨ + ω 2 u = 0,

(3.255)

v¨ + (ω + 2k)v + 2 f NL (v) = 0.

(3.256)

.

2

For this example the equations decouple into a symmetric mode (.x1 = x2 , .v = 0) and a skew-symmetric mode (.x2 = −x1 , .u = 0). A slightly more general situation occurs, if the original equation reads x¨ + ω 2 x1 + f 1 (x1 ) + k(x1 − x2 ) + f 2 (x1 − x2 ) = 0,

(3.257)

x¨2 + ω x2 + f 1 (x2 ) + k(x2 − x1 ) + f 2 (x2 − x1 ) = 0,

(3.258)

. 1

2

where both functions . f 1 and . f 2 are nonlinear. Using again the coordinates u = (x1 + x2 ), v = (x1 − x2 ),

.

(3.259)

we find the equations u¨ + ω 2 u + f 1 ((u + v)/2) + f 1 ((u − v)/2) = 0,

(3.260)

v¨ + (ω + 2k)v + f 1 ((u + v)/2) − f 1 ((u − v)/2) + 2 f 2 (v) = 0.

(3.261)

.

2

The functions .g1 (u, v)

= f 1 ((u + v)/2) + f 1 ((u − v)/2), g2 (u, v) = f 1 ((u + v)/2) − f 1 ((u − v)/2)

(3.262) satisfy g (u, −v) = g1 (u, v),

. 1

g2 (u, −v) = −g2 (u, v).

(3.263)

So .g1 is an even function in .v and .g2 is an odd one. If we assume odd functions for . f 1 and . f 2 , an additional discrete rotational symmetry acts on the system: Rπ : (x1 , x2 ) |→ (−x1 , −x2 ),

.

or (u, v) |→ (−x1 , −x2 ).

(3.264)

3 Normal Forms

95

The system has the symmetry of a rectangle. The action of a continuous family of transformations in a system with 2 DOFs t |→ T (ε, t),

T (0, t) = t,

(3.265)

X i (0, x, t) = xi

(3.266)

.

xi |→ X i (ε, x, t),

can be described as solutions of a system of differential equations ∂T = τ (ε, T ), ∂ε ∂ Xi = ηi (ε, X, T ), ∂ε

T (0, t) = t,

.

X i (0, x, t) = xi ,

(3.267) (3.268)

which are denoted by the vector field ∑ ∂ ∂ ηk . + ∂t ∂xk k=1 n

U=τ

.

(3.269)

To find the action of the transformations on the velocities and accelerations, one needs the prolongations .pr(i) U, which are given by ([5]) U, = pr(1) U = U +

n ∑

.

U,, = pr(2) U = U, +

ζi

k=1 n ∑ k=1

∂ , ∂ x˙k

(3.270)

∂ , ∂ x¨k

(3.271)

δi

with . i

ζ =

∂ηi ∑ ∂ηi ∂τ x˙k − + x˙i , ∂t ∂x ∂t k k=1

(3.272)

δi =

n n ∑ n ∑ ∑ ∂ 2 ηi ∂ηi ∂τ ∂2τ x˙l x˙k + x¨k − 2 x¨i − 2 x˙i . ∂x j ∂xk ∂xk ∂t ∂t j=1 k=1 k=1

(3.273)

n

A system of differential equations . Fi (t, x, x˙ , x¨ ) = 0 is equivariant, if U,, Fi = 0,

.

for all (t, x, x˙ , x¨ ) satisfying F = 0.

(3.274)

If for .n = 2 and .

Fi = m i x¨i +

p k ∑ ∑ k=1 j=0

ck(i)j x1 x2

j k− j

= m i x¨i + f i (x),

(3.275)

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A. Steindl

we select .τ (t) = qt + β, the equivariance conditions read U,, Fi = δi m i + η1

.

= δi m i + η1

p k ∑ ∑

ck(i)j j x1

j−1 k− j x2

+ η2

k ∑

k=1 j=0

∂ Fi ∂ Fi + η2 ∂x1 ∂x2

ck(i)j (k − j)x1 x2

j k− j−1

.

(3.276)

j=0

Zeroing the quadratic velocity terms in .δi yields mi

.

∂ 2 ηi = 0, ∂x12

mi

∂ 2 ηi = 0, ∂x1 ∂x2

mi

∂ 2 ηi = 0, ∂x22

(3.277)

so for .m i /= 0 the .ηi are linear functions in .xi : η = ki1 x1 + ki2 x2 .

. i

(3.278)

Comparison of the acceleration terms in .δi and using the differential equations yields the equations ( mi

.

∂ηi ∂ηi ∂τ x¨1 + x¨2 − 2 x¨i ∂x1 ∂x2 ∂t

) + η1

∂ F1 ∂ F1 + η2 = 0. ∂x1 ∂x2

(3.279)

Using .m i x¨i = − f i we have to solve the equations η f + η2 f 1,2 = k11 f 1 + k12 f 2 − 2q f 1 , η1 f 2,1 + η2 f 2,2 = k21 f 1 + k22 f 2 − 2q f 2 ,

. 1 1,1

(3.280) (3.281)

where . f i,k = ∂ f i /∂xk , or using Lie brackets [η, f ] = −2q f .

.

(3.282)

Equation (3.282) admits two kinds of solutions for special choices of the nonlinearities: • If the nonlinearities are homogeneous polynomials of order .k, the scaling group with generator .q = −(k − 1)/2, η1 = x1 , η2 = x2 (3.283) is a “logical” symmetry: An increase of the variables .xi by the factor .exp(s) can be compensated by a rescaling of time by a factor .exp((1 − k)s/2). • If the nonlinearities admit a symmetry group compatible with the linear contributions, a continuous family of solutions is found.

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97

For example for the rotational invariant potential .V = r 2 /2 + ar 4 /4 with .r 2 = x12 + x22 the Lagrangian equations are x¨ + xi + ar 2 xi = 0, i = 1, 2.

. i

(3.284)

The linear and nonlinear entries satisfy [η, x] = 0, [η, ar 2 x] = 0, for η1 = −x2 , η2 = x1 .

.

(3.285)

So any direction .x2 = cx1 would be a possible Nonlinear Normal Mode.

3.3.4 Discrete Symmetries and Circular Systems Nonlinear Normal Modes frequently show localization behaviour, the motion in the system is concentrated on a small domain. Similar observations hold for symmetric systems, where the localization is caused by the symmetry of different solution branches. As an example we have a look at a symmetric system, for which the equivariant bifurcation theory ([3]) predicts the shape of the observed oscillations: We consider a system of .k identical masses with cyclic symmetry and nearestneighbour coupling displayed in Figs. 3.10 and 3.11. x¨ = − f (u) (xi ) − fˆ(c) (xi+1 − xi ) − fˆ(c) (xi−1 − xi ).

. i

(3.286)

This system admits the dihedral group .Dk of cyclic permutations .xi |→ xi+1 and mirror reflections, e.g. .x1 ↔ x2 . In the space.(x1 , x2 , x3 ) the group operations are represented by the.3 × 3 matrices

Fig. 3.10 Simple system of three identical bodies with 3-fold rotational symmetry

98

A. Steindl

⎛

1 .e = ⎝0 0 ⎛ 0 S12 = ⎝1 0

⎞ 0 0⎠ , 1 ⎞ 10 0 0⎠ , 01

⎛

0 1 0

0 R 1 = ⎝1 0 ⎛ 1 S23 = ⎝0 0

⎞ 1 0⎠ , 0 ⎞ 00 0 1⎠ , 10

⎛ 0 R2 = ⎝0 1 ⎛ 0 S31 = ⎝0 1

0 0 1

⎞ 0 1⎠ , 0 ⎞ 01 1 0⎠ , 00 1 0 0

(3.287)

(3.288)

with e.g. . R12 = R2 = R1−1 , . R1 S12 = S13 , and . R1 S13 = S23 . By introducing a new basis . x = B y with √ √ ⎞ ⎛ √ 1/√3 −1/√6 −1/√ 2 .B = ⎝1/ 3 −1/ 6 1/ 2 ⎠ , (3.289) √ √ 1/ 3 2/3 0 the matrices for . R1 and . S12 become ⎛ ⎞ 1 0 0 . R1 = ⎝0 cos ϑ sin ϑ ⎠ , 0 − sin ϑ cos ϑ

⎛

S12

⎞ 10 0 = ⎝0 1 0 ⎠ , 0 0 −1

(3.290)

with .ϑ = 2π/3. In the new basis the actions of the group have been simultaneously block-diagonalized: All rotations and reflections keep . y1 fixed and act as 2dimensional maps on . y2 and . y3 . Introducing complex coordinates .z = y2 + i y3 and . z = y2 − i y3 , the symmetry operations are given by .

R1 : z |→ exp(iϑ)z, z |→ exp(−iϑ)z,

S12 : z ↔ z.

(3.291)

According to [3] the symmetry restricts the possible linear and non-linear systems: A .3 × 3 matrix .A, which commutes with the group operations, that is AR = RA, for R ∈ {R1 , S12 },

(3.292)

A = diag(a, b, b).

(3.293)

.

has the structure .

The synchronous motion . y1 /= 0 and the interacting modes .(y2 , y3 ) /= 0 decouple linearly. And double eigenvalues occur generically. Nonlinear equivariant functions . f j (y1 , z, z) satisfy the relations

3 Normal Forms

99

f (y1 , exp(iϑ)z, exp(−iϑ)z) = f 1 (y1 , z, z), f 1 (y1 , z, z) = f 1 (y1 , z, z), (3.294)

. 1

f 2 (y1 , exp(iϑ)z, exp(−iϑ)z) = exp(iϑ) f 2 (y1 , z, z), f 2 (y1 , z, z) = f 3 (y1 , z, z), (3.295) f 3 (y1 , exp(iϑ)z, exp(−iϑ)z) = exp(−iϑ) f 3 (y1 , z, z), f 3 (y1 , z, z) = f 2 (y1 , z, z). (3.296) Representing . f j in power series . f j = to satisfy the conditions m2 − m3 ≡ 0

.

∑ m

f j,m y1m 1 z m 2 z m 3 , the exponents .m k have

(mod 3)

m 2 − m 3 ≡ 1 (mod 3) m 2 − m 3 ≡ −1 (mod 3)

for j = 1,

(3.297)

for j = 2, for j = 3.

(3.298) (3.299)

Assuming . f j (0) = 0 and linear decoupling, we obtain the leading terms due to symmetry f = ay1 + f 1,2,0,0 y12 + f 1,0,1,1 zz + f 1,3,0,0 y13 + f 1,1,1,1 y1 zz + f 1,1,3,0 (z 3 + z 3 ), (3.300)

. 1

f 2 = bz + f 2,0,0,2 z 2 + f 2,1,0,2 y1 z 2 + f 2,0,2,1 z 2 z.

(3.301)

3.3.5 Hopf Bifurcations for Circular Systems We consider a symmetric arrangement of . N ≥ 3 identical systems with nearestneighbour coupling; the systems are non-conservative and we assume that a Hopf bifurcation (a complex conjugate pair of eigenvalues crosses the imaginary axis) occurs at the undeformed configuration. x˙ = f (u) (x i ) + f (c) (x i+1 − x i ) + f (c) (x i−1 − x i ),

. i

(3.302)

where . f (u) (x i ) governs the internal dynamics of the .ith subsystem and . f (c) denotes the couplings between adjacent masses (Fig. 3.11). With .A = ∂ f (u) (x)/∂ x and .B = ∂ f (c) (u)/∂u the linearized equations for the .ith system are ˙ i = Bx i−1 + (A − 2B)x i + Bx i+1 . .x (3.303) We assume that .B /= 0. For . N = 3 and . N = 4 the system matrices have the structure

100

A. Steindl

Fig. 3.11 System with the symmetry of a square

C(4)

⎞ ⎛ A − 2B B B (3) A − 2B B ⎠, .C =⎝ B B B A − 2B ⎛ ⎞ A − 2B B 0 B ⎜ B A − 2B B 0 ⎟ ⎟. =⎜ ⎝ 0 B A − 2B B ⎠ B 0 B A − 2B

(3.304)

(3.305)

The system has the dihedral symmetry .D N , which acts by cyclic permutations x |→ x i+1 and reflections . x i |→ x −i (mod N ) . By this symmetry the matrices can be put in block-diagonal form. Let .ζ ∈ C be an . N th root of unity: .ζ N = 1, set

. i

.

u = (x, ζ x, ζ 2 x, . . . , ζ N −1 x)T ,

(3.306)

then Cu = ( y, ζ y, ζ 2 y, . . . , ζ N −1 y)T with y = (A − 2B + ζB + ζB)x.

.

(3.307)

If . x is an eigenvector of .Cζ = A − 2B + ζB + ζB, then .u is an eigenvector of .C. It is therefore sufficient, to calculate the eigenvectors of the matrices .Cζ . There are 3 different cases: • .ζ = 1: .C1 = A, the synchronous modes, • .ζ = −1 (for even. N ).C−1 = A − 4B; neighbouring systems oscillate in anti-phase • .Im ζ /= 0: There are 2 identical matrices .Cζ = Cζ = A + (2Kζ − 2)B. All eigenvalues of .Cζ occur twice in .C. We assume that for some .ζ ∈ / {−1, 1} a pair of eigenvalues of .Cζ becomes purely imaginary. Then due to the symmetry the critical eigenspace is 4-dimensional. Introducing complex coordinates for the two modes, the equations of motion at the Hopf bifurcation point become

3 Normal Forms

101

z˙ = iωz 1 + f 1 (z 1 , z 1 , z 2 , z 2 ),

. 1

z˙ 2 = iωz 2 + f 2 (z 1 , z 1 , z 2 , z 2 ).

(3.308)

The circular symmetry acts by . 1

z |→ ζz 1 ,

z 1 |→ ζz 1 ,

(3.309)

z 2 |→ ζz 2 ,

z 2 |→ ζz 2 .

(3.310)

The reflectional symmetry acts by.z 1 ↔ z 2 . The functions. f i (z 1 , z 1 , z 2 , z 2 ) satisfy f (ζz 1 , ζz 1 , ζz 2 , ζz 2 ) = ζ f 1 (z 1 , z 1 , z 2 , z 2 ),

(3.311)

f 2 (ζz 1 , ζz 1 , ζz 2 , ζz 2 ) = ζ f 2 (z 1 , z 1 , z 2 , z 2 ),

(3.312)

. 1

f 1 (z 2 , z 2 , z 1 , z 1 ) = f 2 (z 1 , z 1 , z 2 , z 2 ).

(3.313)

Applying Normal Form theory introduces an extra phase-shift symmetry: f (eiωt z 1 , e−iωt z 1 , eiωt z 2 , e−iωt z 2 ) = eiωt f i (z 1 , z 1 , z 2 , z 2 ).

. i

(3.314)

With .ζ = exp(kiϑ), .ϑ = 2π/N and .k ∈ {1, . . . , N − 1}, .k /= N /2 we obtain the conditions for the monomials . z m in the power series expansion of . f 1 : k(m 1 − m 2 − m 3 + m 4 ) ≡ k (mod N ), m 1 − m 2 + m 3 − m 4 = 1.

.

(3.315) (3.316)

With .q = gcd(k, N ) the first condition (3.315) becomes m1 − m2 − m3 + m4 ≡ 1

.

(mod q).

(3.317)

For . N → ∞ the discrete symmetry .D N passes to the continuous symmetry .O(2) of arbitrary rotations. The Normal Form conditions become .

m 1 − m 2 − m 3 + m 4 = 1,

(3.318)

m 1 − m 2 + m 3 − m 4 = 1,

(3.319)

with solutions .m 1 − m 2 = 1, .m 3 − m 4 = 0. Up to third order the bifurcation equations are z˙ = (λ + iω + A1 |r1 |2 + A2 |r2 |2 )z 1 ,

(3.320)

z˙ 2 = (λ + iω + A2 |r1 | + A1 |r2 | )z 2 ,

(3.321)

. 1

2

with . A j = c j + id j . The amplitude equations read

2

102

A. Steindl

r˙ = (λ + c1 |r1 |2 + c2 |r2 |2 )r1 ,

(3.322)

r˙2 = (λ + c2 |r1 | + c1 |r2 | )r2 .

(3.323)

. 1

2

2

These equations admit 2 different non-trivial types of solution: • Rotating wave: .r1 /= 0, .r2 = 0. The solution is supercritical, if .c1 < 0; it is stable, if .c1 < 0 and .c2 − c1 < 0. • Standing wave: .r2 = r1 /= 0. The solution is supercritical, if .c1 + c2 < 0; it is stable, if .c1 + c2 < 0 and .c1 − c2 < 0. For . N = 4, the Normal Form conditions m1 − m2 − m3 + m4 ≡ 1

.

(mod 4),

(3.324)

m1 − m2 + m3 − m4 = 1

(3.325)

admit the solutions .m 1 − m 2 = 1 + 2k, .m 3 − m 4 = −2k with .k ∈ Z. For .k = 0 we obtain the entries for the .O(2)-symmetric case. For .k = −1 we find the additional solution .m 2 = 1, .m 3 = 2. Up to third order the bifurcation equations read z˙ = (λ + iω + A1 |r1 |2 + A2 |r2 |2 )z 1 + A3 z 1 z 22 ,

(3.326)

z˙ 2 = (λ + iω + A2 |r1 | + A1 |r2 | )z 2 +

(3.327)

. 1

2

2

A3 z 12 z 2 .

These admit three different types of steady oscillations: • Rotating wave .z 1 /= 0, .z 2 = 0. It is supercritical if .c1 < 0 and stable if .c1 < 0, .c2 − c1 < 0 and .| A2 − A1 |2 − |A3 |2 > 0. • Standing wave 1: .z 2 = z 1 /= 0. It is supercritical, if .c1 + c2 + c3 < 0 and stable, if .c1 + c2 + c3 < 0, .c1 − c2 − 3c3 < 0 and .|A3 |2 − K((A1 − A2 )A3 ) > 0. • Standing wave 2: .z 2 = i z 1 /= 0. It is supercritical, if .c1 + c2 − c3 < 0 and stable, if .c1 + c2 − c3 < 0, .c1 − c2 + 3c3 < 0 and .|A3 |2 + K((A1 − A2 )A3 ) > 0. The oscillation pattern for the standing waves is shown in Fig. 3.12. J. W. Swift [8] observed that also asymmetric solutions occur generically for .D4 -symmetric Hopf bifurcations. By introducing spherical coordinates u + iv = r sin θ exp(iϕ)

.

w = r cos θ

= 2z 1 z 2 ,

(3.328)

= |z 1 | − |z 2 | , 2

2

(3.329)

the interesting dynamics can be studied on the surface of a sphere (Fig. 3.13a). Besides the mentioned symmetric solution types also an asymmetric oscillation and a modulated rotating wave T are observed generically.

3 Normal Forms

103

v

v v

v

v

v

Fig. 3.12 Different kinds of Standing waves for a problem with the symmetry of the square. For the standing wave 1 (left) the systems . x 2 and . x 4 are at rest, while . x 1 and . x 3 oscillate synchronously. For the standing wave 2 (right) the configuration remains symmetric w.r.t. the diagonal

Fig. 3.13 Dynamics on the sphere and possible bifurcation diagram for a system with the 4-fold symmetry .D4

Summary for Hopf bifurcations with Circular Symmetries • Circular symmetries frequenctly cause double eigenvalues, • Bifurcation equations yield several kinds of synchronous solutions. Also rotating waves occur, which are not considered by NNMs. • For special cases (e.g. .D4 ) even quasi-periodic periodic motions and asymmetric branches bifurcate generically. • No special restrictions (e.g. conservative systems) on the bifurcation equations, except symmetry.

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A. Steindl

3.4 Conclusions In the first two sections a short introduction into the method of Normal Forms for general and Hamiltonian systems has been presented. Normal Forms allow to reduce the given system to a much simpler form and derive solutions. In the third section the main ideas for Nonlinear Normal modes were introduced; it was explained, how these solutions can be calculated approximately and how these modes are related to symmetry properties of the underlying system. Further the treatment of a rotational symmetric system, which is frequently analyzed using the NNM method, can be investigated using equivariant bifurcation theory, yielding the list of expected solution types in a systematic way. The author is convinced that the interaction between equivariance and NNMs promises a deeper understanding of the observed phenomena.

References 1. Arnold, V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations. SpringerVerlag, New York – Heidelberg – Berlin (1983) 2. Elphick, C., Tirapegui, E., Brachet, M., Coullet, P.: A simple global characterization for normal forms of singular vector fields. Physica 29D pp. 95–127 (1987) 3. Golubitsky, M., Stewart, I., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory. No. 69 in Applied Mathematical Sciences. Springer-Verlag, New York (1988) 4. Meyer, K.R., Offin, D.C.: Introduction to Hamiltonian Dynamical Systems and the . N -Body Problem. No. 90 in Applied Mathematical Sciences. Springer-Verlag (2017) 5. Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer-Verlag (1986) 6. Rosenberg, R.: On nonlinear vibrations of systems with many degrees of freedom. In: G. Chernyi, H. Dryden, P. Germain, L. Howarth, W. Olszak, W. Prager, R. Probstein, H. Ziegler (eds.) Advances in Applied Mechanics, Advances in Applied Mechanics, vol. 9, pp. 155 – 242. Elsevier (1966). https://doi.org/10.1016/S0065-2156(08)70008-5. URL http://www. sciencedirect.com/science/article/pii/S0065215608700085 7. Shaw, S., Pierre, C.: Normal modes of vibration for non-linear continuous systems. Journal of Sound and Vibration 169(3), 319 – 347 (1994). https://doi.org/10.1006/jsvi.1994.1021. URL http://www.sciencedirect.com/science/article/pii/S0022460X84710212 8. Swift, J.W.: Hopf bifurcation with the symmetry of the square. Nonlinearity 1, 333–377 (1988) 9. Vakakis, A., Manevitch, L.I., Mikhlin, Y.V., , Pilipchuk, V.N., Zevin, A.A.: Normal modes and localization in nonlinear systems. Wiley, New York (1996) 10. Wiggins, S.: Introduction to applied nonlinear dynamical systems and chaos. Springer, New York (2003)

Chapter 4

Nonlinear Normal Modes and Reduced Order Models Paulo Batista Gonçalves

Abstract Nonlinear normal modes (NNMs) have been used in the last decades as an additional tool for the analysis of multi degree of freedom (mdof) dynamical systems and for the derivation of reduced order models. Examples in literature show that only a few NNMs are usually required to model, in many problems, their forced dynamics. In this chapter a brief literature review with the seminal contributions in this field and recent contributions are presented. Then the definitions of NNMs based on the works by Rosenberg and Shaw and Pierre are given. According to the latter definition, a nonlinear vibration mode is a two-dimensional invariant manifold in the phase space of the system. This is approximated by asymptotic expansions in terms of a pair of master coordinates or through a Galerkin expansion. This methodology and additional numerical tools are presented here together with didactical examples that illustrate some unique characteristics of NNMs as compared to linear normal modes. Also Poincaré sections are used for the identification of NNMs and their stability and continuation methods are used to obtain frequency-energy plots and frequency-amplitude relations. The concept of multi-mode is briefly presented.

4.1 Introduction The superposition theorem is the cornerstone of linear systems theory [6]. This allows the classical modal analysis of linear dynamical systems based on the concept of linear vibration modes [19]. Consider an .n degrees of freedom linear system. The linear modal analysis of the system leads to the determination of .n eigenvalues (frequencies) and .n eigenvectors (vibration modes): .

{x} ¨ = [A] {x} ,

(4.1)

where matrix . A depends only on the system parameters. P. B. Gonçalves (B) Pontifical Catholic University of Rio de Janeiro, PUCRio, Rio de Janeiro, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. R. Castilho Piqueira et al. (eds.), Lectures on Nonlinear Dynamics, Understanding Complex Systems, https://doi.org/10.1007/978-3-031-45101-0_4

105

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P. B. Gonçalves

Using this information, the coupled system can be transformed into a system .n decoupled second order differential equations of motions in terms of the so called modal coordinates [6]. Solving these equations the response of the free and forced vibrations can be obtained and, by an inverse problem, the response can be finally obtained in terms of the real coordinates. Also, in linear systems, by the principle of superposition, a large-scale model can be reduced to a reduced order model, represented by the dominant vibration modes, using the usual modal analysis tools, as the modes are linearly independent. Modal analysis is widely used in engineering practice since many structures oscillates in a linear regime. Damping is usually considered through a Rayleigh damping which is proportional to a linear combination of mass and stiffness. However, for nonlinear systems, these tools are not directly applicable and other ways must be used to understand and analyze the dynamics of such systems. Due to recent technological advances, an increasing number of engineering systems exhibits some type of nonlinearity, thus a nonlinear analysis becomes necessary. Common sources of nonlinearity include geometric nonlinearities (present in slender structural systems), material nonlinearities (nonlinear constitutive law), inertial nonlinearities, nonlinear boundary conditions (equation unilateral boundary conditions), some displacement and velocity dependent forces such as hydrodynamic, aeroelastic and electromagnetic forces, friction, etc. Nonlinear systems can exhibit extremely complex behaviors which linear systems cannot. These phenomena include dynamic jumps, bifurcations such as flip, pitchfork and Hopf, subharmonic, superharmonic and internal resonances, saturation phenomena, limit cycles, modal interaction and chaos, to name a few. Analytical (exact or approximate) and numerical methods must be used to obtain Poincaré maps, bifurcation diagrams, frequency–energy plots, backbone curves and basins of attraction, among others. Nonlinear normal modes (NNMs) have been proposed as a tool for the analysis of nonlinear dynamical systems and the derivation of reliable reduced order models, using some of the ideas of the linear modal analysis. In particular, due to the invariance of the nonlinear normal modes, the number of simulated modes can be reduced efficiently.

4.2 Brief Literature Review NNMs were pioneered in the 1960s thanks to the seminal work of Rosenberg [35, 36]. Following the works by Rosenberg in the early 1970s, Rand, Manevitch and Mikhlin explored the concept of NNMs in the analysis of mdof discrete systems [17, 33]. Their work was expanded by Vakakis in his doctoral dissertation in 1990 [47] which was followed by several works by Vakakis and his co-workers [48]. Shaw and Pierre, also in the early 1990s, developed a more generalized definition of NNMs based on the theory of invariant manifolds and applied it to nonlinear discrete and continuous system [37, 39]. In 2002, Pesheck et al. [32] developed a Galerkin-based approach for the nonlinear normal modes through invariant manifolds. Review works and applications in structural dynamics are reported in [3, 46]. The derivation of

4 Nonlinear Normal Modes and Reduced Order Models

107

reduced order models (including in some cases the influence of non-simulated modes) through nonlinear normal modes is a promising research line since it enables the use of most analytical and geometrical tools used in the analysis of nonlinear dynamical systems [43]. Peeters et al. [30] propose the use of continuation techniques toward a practical computation of NNMs, while the use of the finite-element method in computing NNM and system reduction can be found in [2, 18, 41]. The phenomenon of internal resonance, such as .1:1, .1:2 and .1:3, observed in many structural and mechanical systems, can be analyzed using coupled nonlinear normal modes. The application of NNMs to problems involving internal resonances is studied Nayfeh et al. [27] and Lacarbonara et al. [16] using the method of multiple scales as a reduction technique. Considering these problems, Pesheck et al. [31] proposed an extension of the invariant manifold approach for deriving nonlinear normal modes of vibration to the case of multi-mode manifolds. According to Pesheck et al. [31], the dynamic models obtained using multi-mode manifolds can capture the essential coupling and interaction between the modes of interest, while avoiding coupling from other non-essential modes. Recently Meyrand et al. [20] proposed a continuation method combined to a model reduction based on Proper Generalized Decomposition (PGD) technique, mixing PGD, harmonic balance method (HBM) and basic continuation techniques in order to reach a highly reduced description of the system.

4.3 Definition of NNMs An .n degrees of freedom discrete autonomous conservative system can be described by a set of .n equations of motion in the form .

{ } ( ) x¨i = f i x1 , x2 , . . . , xn , i = 1, 2, . . . , n,

(4.2)

where .xi represent the generalized coordinates measured from an equilibrium reference state and . f i are nonlinear functions of these coordinates and the overdot denotes time derivatives. The concept presented by Rosenberg [35, 36] of nonlinear modes is considered as an extension of linear vibration modes, and has become a useful tool in the analysis of nonlinear vibrations. According to Rosenberg, NNMs are defined as periodic, synchronous motions (vibration in unison) represented by defined relations among the generalized coordinates (in the linear case, the components of each eigenvector). All generalized coordinates perform periodic motions (not necessarily harmonic) and have the same period of vibration and passes through the equilibrium position and reach the maximum displacement simultaneously (as in the linear case). Thus, for a given mode, the oscillations of all coordinates can be parameterized by a single coordinate leading to a SDOF system for each nonlinear mode. The possibility of defining the positions of all the masses through any one of them allows a very efficient order reduction for the problem. For a given .r ∈ [1, 2, . . . , n], at any time .t the system coordinates must be related by functional equations of the form

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x = Pi (xr ), i = 1, 2, . . . , n, i /= r,

. i

(4.3)

where . Pi is called a modal function for the nonlinear mode .r . The trajectories of the system in configuration space corresponding to the normal modes are called modal lines. If the modal lines are straight lines, they are called similar modes. When a system moves in a similar mode, the motion occurs along a straight line passing through the equilibrium position in the system .n-dimensional configuration space. The linear mode can, by this definition, be seen as a particular case of the nonlinear mode, where the modal functions in Eq. (4.3) are linear functions. The motion in a similar mode in a free vibrating system can be mathematically written as: . x i = cir xr , i = 1, 2, . . . , n, i /= r, crr = 1. (4.4) These linear relations must be satisfied by the coordinates .xi for all times, where c are .n − 1 unknown scalar quantities. If they are described by nonlinear relations (curves), they are called non-similar modes. While similar modes can be found in linear and nonlinear systems, nonsimilar modes are found only in nonlinear systems. The existence of similar normal modes in nonlinear systems can be connected to symmetries of the physical system which are reflected in symmetry properties in its expression of the potential energy and associated equations of motion [29]. By its very nature, Rosenberg’s definition has two limitations: (I) it cannot be applied to non-conservative systems and (II) cannot be used when the problem presents internal resonances. These limitations stem from the fact that in both cases the system vibration is not synchronous. In this context the definition of Shaw and Pierre [37] is an alternative for the extension of the concepts involving the nonlinear normal modes for systems with non-unison vibration. This original concept of nonlinear modes was modified by Shaw and Pierre [37] who proposed a definition of nonlinear modes, in which not only the generalized displacements, but also the generalized velocities must be considered. According to them, a nonlinear mode is an invariant two-dimensional manifold immersed in the phase space of the system. Since manifolds are invariant, this means that if the initial conditions are in a given manifold, the corresponding motion remains in that manifold. The advantage of this definition is that it incorporates Rosenberg’s definition as a particular case and is suitable for both conservative and non-conservative systems. In the linear case, the manifold is simply the plane associated with the linear vibration mode. Formally these concepts can be expressed by: an invariant manifold of a dynamical system is a subset . S of the phase space, such that, if a set of initial conditions is given in . S, the solution remains in . S over all time. This manifold passes through the equilibrium point of interest and, at that point, is tangent to the two-dimensional planar manifold of the linearized system [40]. In this formulation, a pair of coordinates, displacement-velocity, say .(u, v), are chosen as governing or master coordinates, characterizing the individual nonlinear modal motion and all the remaining degrees of freedom, called slave coordinates, are described as dependent coordinates, composed in the same way by pairs of displacement-velocity coordi-

. ir

4 Nonlinear Normal Modes and Reduced Order Models

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nates. Thus, in this manifold, the dynamic system is governed by an equation of motion involving only a pair of state variables, that is, it behaves as a single degree of freedom nonlinear system. One feature of NNMs not found in linear systems, is that their number can be higher than the number of degrees of freedom (superabundance of modes) of the discrete model due to system symmetries, internal resonances and modal coupling. Also, additional nonlinear modes can be obtained, as the system total energy increases, due to mode bifurcations, and in such case, they cannot be regarded as continuation of the underlying LNMs. While in the linear case all modes are neutrally stable, NNMs can be stable or unstable. This is another characteristic of NNMs that differentiates them from the linear case. In agreement with Lyapunov stability concept, instability means that small perturbations on the initial conditions that generate the NNM motion lead to the elimination of the mode oscillation [13]. The stability may be computed, for example, numerically through the eigenvalues of the monodromy matrix, analytically through Floquet theory after adding small perturbations to the periodic solutions or through appropriate Poincaré sections.

4.4 Use of Poincaré Sections to Identify NNMs According to Vakakis [47], the use of Poincaré sections is an efficient tool to determine numerically the existence of NNMs, especially in the case where mode multiplicity occurs. Techniques for obtaining the Poincaré maps of two-dimensional systems are presented in [22]. For a 2dof conservative structural system where the Lagrangian is given by . L = T − Π , where . T is a quadratic function of the velocities and .Π is the total potential energy, the Hamiltonian, . H , is the total energy of the system, that is, . H = 2T − (T − H ) = T + Π . Fixing the energy level, one can restrict the flow of the dynamical system to a three-dimensional isoenergetic boundary. This is achieved by setting . H = h, where .h is the assumed energy level. If another independent first integral exists, then the energy manifold . H = h is fibered by invariant twodimensional tori [49]. If the three-dimensional isoenergetic region is cut by a 2D plane and the flow is transverse to this plane [49], the resulting cross section .Σ is two-dimensional and defines the Poincaré map. For example, consider the cut plane defined by .Γ = qi = 0, where .qi is a generalized coordinates, the Poincaré section for an energy level . H = h is given by .Σ = {qi = 0, q˙i > 0} ∩ {H = h}, .i = 1, 2 [22]. The restriction on the sign of the velocity is because the Poincaré section must be orientation preserving [10, 22]. A motion beginning on .Σ returns to .Σ after making a circuit around the torus, producing a mapping of .Σ onto itself. When the energy level increases, as in most nonlinear Hamiltonian systems, the complexity increases due to bifurcations of NNMs and appearance of the stochastic layers associated to chaotic motions [10]. Using these techniques, one can observe the global flow of the dynamical system close to each mode, and thus identify the nonlinear modes and characterize their

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Fig. 4.1 Poincaré section of the 2dof Augusti model displaying superabundance of modes due to internal resonance and system symmetries [28, 29]

stability. Stationary modes correspond to stationary points in the Poincaré section. If the stationary point is surrounded by a closed curve, the mode is stable, while the curve corresponds to a stable quasi-periodic motion of the Hamiltonian system. On the other hand, if the neighborhood looks like a saddle, the mode is unstable. In short, they are stable if they are centers (elliptical fixed points) and unstable if they are saddles (hyperbolic points). Any other point on the Poincaré section corresponds to a time response with varying amplitude and phase. Thus in order to obtain the Poincaré section, one must follows the steps: 1. Define the desired 2D Poincaré section, e.q., .q1 × q˙1 ; 2. Define the window of initial conditions based on the conservation of energy principle, since it delimits the Poincaré section region of interest (maximal equipotential surface, modal line perpendicular to the maximal surface, .Π + T = h); 3. Specify one additional variable as zero, e.q., .q2 = 0; 4. Obtained the remaining variable for the specified energy level .h. For example, .Π + T = h ⇒ solve for .q ˙2 ; 5. Discretize the chosen phase plane into a sufficiently large number of cells; 6. Using each cell coordinates as initial conditions; integrate the nonlinear equations of motion for a defined time length; 7. During this integration, the section coordinates (here points .q1 and .q˙1 ) are saved every time that the following conditions are satisfied.q2 = 0,.q˙2 > 0(unidirectional crossing). Orlando et al. [28, 29] studied in detail the nonlinear dynamics of the wellknown 2dof Augusti’s model where both static and dynamic nonlinear interactions are observed. Figure 4.1 illustrates a typical Poincaré section of the system where five stable NNMs and two unstable nonlinear modes are observed.

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4.5 Asymptotic Method For the development of the methodology, the equations of motion of an autonomous mdof dynamical system can be written as a system of first order equations: x = y, y˙ = f (x, y) ,

(4.5)

.

where .xT = [x1 , . . . , xn ]T and .yT = [x˙1 , . . . , x˙n ]T = [y1 , . . . , yn ]T are respectively the generalized displacements and velocities and.f T = [ f 1 , . . . , f n ]T are position and velocity dependent nonlinear functions. The procedure then begins with the hypothesis that at least one system motion can be obtained in which all displacements and velocities are related by means of nonlinear functions to a single pair displacement-velocity, chosen arbitrarily. Here, for the sake of simplicity, the first degree-of-freedom displacement-velocity pair, .x1 e . y1 , is chosen as the master pair [37–39]. Also, in order to simplify the notation u = x1 , v = y1 .

(4.6)

.

This pair of variables describes completely the invariant manifold for the chosen nonlinear mode. The other slave pairs, corresponding to the other degrees of freedom, are represented in terms of .u and .v by the constraint functions .(Pi , Q i ) as follows: x = Pi (u, v), v = Q i (u, v), i = 1, 2, 3, . . . , n,

. i

(4.7)

where . P1 (u, v) = u and . Q 1 (u, v) = v. The explicit time dependence of the equations can be eliminated by the time derivatives of the constraint equations, using the chain rule: ∂ Pi (u, v) ∂ Pi (u, v) u˙ + v, ˙ i = 1, 2, 3, . . . , n, ∂u ∂v ∂ Q i (u, v) ∂ Q i (u, v) y˙i = u˙ + v, ˙ i = 1, 2, 3, . . . , n. ∂u ∂v x˙ =

. i

(4.8) (4.9)

By replacing Eqs. (4.8) and (4.9) into the equations of motion (4.5) and by the use of the master pair definitions (4.6) and the slave relations (4.7), the following system of .2n − 2 second order partial differential equations (PDEs) is obtained: . Q i(u,v)=

∂Pi(u,v) ∂Pi(u,v) v+ f 1(u,P2(u,v),. . . ,Pn(u,v);v,Q 2(u,v),. . . ,Q n(u,v)), ∂u ∂v

(4.10)

.

f i (u, P2 (u, v), . . . , Pn (u, v); v, Q 2 (u, v), . . . , Q n (u, v)) =

∂ Q i (u, v) v+ ∂u

∂ Q i (u, v) f i(u,P2(u,v),. . . ,Pn(u,v);v,Q 2(u,v),. . . ,Q n(u,v)), i = 2,3,. . . ,n. (4.11) ∂v

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Once the. Pi and. Q i functions are obtained by solving these PDEs, the dynamics on the invariant manifold subspace is reduced to a single degree-of-freedom nonlinear modal oscillator, governed by the master pair of coordinates. It is simply obtained by replacing these functions in in the first pair of equations of motion, resulting in: u˙ = v, v˙ = f 1(u,P2(u,v),. . . ,Pn(u,v);v,Q 2(u,v),. . . ,Q n(u,v)).

.

(4.12)

In general, there is no exact closed form solution to the partial differential equations that govern the invariant manifold. The approximate procedure used here is based on the assumption that the system nonlinear normal modes can be expressed as a Taylor series around the equilibrium configuration, taken here, for simplicity as .x0 = 0. Thus the constraint functions can be written as: . Pi(u,v) = a1i u +a2i v+a3i u

+a4i uv+a5i v 2+a6i u 3+a7i u 2 v+a8i uv 2+a9i v 3+· · · ,

2

i = 1, 2, . . . , n,

. Q i(u,v) = b1i u +b2i v+b3i u

(4.13)

+b4i uv+b5i v 2+b6i u 3+b7i u 2 v+b8i uv 2+b9i v 3+· · · ,

2

i = 1, 2, . . . , n.

(4.14)

This approach for the approximate solution of the system of differential Eqs. (4.10) and (4.11) is local by nature, and results in approximations to the normal modes of vibration and dynamics close to the equilibrium point. These approximations are expressed in terms of power series and can, in principle, be generated for systems of any order [38]. These equations can be written in a vector form as: ⎧ ⎫ ⎛⎡ 1 x1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎜⎢ 0 ⎪ ⎪ y ⎪ ⎪ 1 ⎜⎢ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎜⎢a12 ⎪ ⎨ x2 ⎪ ⎬ ⎜⎢ ⎢ y2 = ⎜ . ⎜⎢b12 ⎢ . ⎜ ⎪ ⎪ . ⎪ ⎪ .. ⎪ ⎪ ⎜⎢ .. ⎪ ⎪ ⎢ ⎜ ⎪ ⎪ ⎪ ⎪x ⎪ ⎪ ⎝⎣a ⎪ n⎪ 1n ⎪ ⎪ ⎩ ⎭ yn b1n ⎡

⎤ ⎡ 0 1 ⎢ 1⎥ 0 ⎥ ⎢ ⎢ a22 ⎥ ⎥ ⎢a32 u + a42 v ⎥ b22 ⎥ + ⎢ ⎢b32 u + b42 v .. ⎥ ⎢ .. ⎢ . ⎥ . ⎥ ⎢ a2n ⎦ ⎣a3n u + a4n v b2n b3n u + b4n v

1 ⎢ 0 ⎢ ⎢a62 u 2 + a82 v 2 ⎢ 2 2 ⎢ + ⎢b62 u + b82 v ⎢ .. ⎢ . ⎢ ⎣a u 2 + a v 2 6n 8n b6n u 2 + b8n v 2

⎤⎞ 0 ⎟ 1 ⎥ ⎥⎟ ⎥ a52 v ⎥⎟ ⎟{ } ⎟ u b52 v ⎥ + ⎥⎟ .. ⎥⎟ v ⎟ . ⎥ ⎥⎟ a5n v ⎦⎠ b5n v ⎤

0 ⎥ 1 ⎥ a72 u 2 + a92 v 2 ⎥ ⎥{ } u b72 u 2 + b92 v 2 ⎥ + ··· , ⎥ ⎥ v .. ⎥ . ⎥ a7n u 2 + a9n v 2 ⎦ b7n u 2 + b9n v 2

(4.15)

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or more compactly: ⎧ ⎫ x1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ y1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x2 ⎪ ⎪ { } ⎬ ([ ] [ ⎨ ⎪ ] [ ]) u y 2 = μ0 + μ1 (u, v) + μ2 (u, v) . {z} = + ··· , v ⎪ .. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪.⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪xn ⎪ ⎪ ⎭ ⎩ ⎪ yn

(4.16)

where .μ0 , .μ1 and .μ2 are rectangular matrices of dimension .2n×2. Matrix .μ0 is the modal linear component, while .μ1 and .μ2 represent the effects of quadratic and cubic nonlinear terms, respectively. The nonlinear terms of Eq. (4.16) describe the curvature of the modal subspace. These terms capture the effects of the nonlinearities and result in the fact that the ratios between velocities and displacements depend on the range of motion. The substitution of Eqs. (4.13) and (4.14) into the equation of motion and coefficients in terms of the same degree are equated resulting in a system of algebraic equations in terms of the constraint equation coefficients .ai j and .bi j , which can be solved sequentially. The main limitation of this approach is that it is only locally valid [27]. The region of validity, as in any asymptotical method, is not known a priori, being determined only by comparison with numerical solutions of the original problem or by convergence analyses. For example, for a system with expansion up to cubic terms, this leads to a system of .18(n − 1) equations. These equations can be solved sequentially, as in any perturbation method. There is no theoretical limitation to the asymptotic developments. One may increase the expansion order, but it be-comes more and more difficult and cumbersome and may lead to erroneous results. The equations for linear terms are is equivalent to solving an eigenvalue problem, so traditional methods of linear algebra can be used. The equations of quadratic coefficients depend on the solution of the coefficients of linear terms, and therefore result in linear systems of the unknowns, usually having a single solution for each mode and so on. These characteristics make it possible for the method to be easily used in computational solution schemes, even for systems with many degrees of freedom [2, 18, 41]. With these coefficients, the approximate local solution for the dynamic equations of the system can be obtained for each mode. This procedure results in an equation of a nonlinear oscillator of a single degree of freedom that represents the dynamics of the system in a two-dimensional subspace (invariant manifold) that is tangent to the auto-space of the linear normal mode at the equilibrium point.

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4.6 Galerkin Based Procedure One of the basic differences between the asymptotic methods and the one based on the Galerkin procedure, as proposed by Pesheck et al. [32], is the use of polar coordinates in the geometric determination of the invariant manifolds. The differential equations of the invariant manifolds are solved in an approximate way, using a discretization by means of a Galerkin projection over a previously chosen range and phase angle domain [32]. The transformation for the polar coordinate system can be defined as a function of time dependence on the amplitude .a and the phase angle .φ as: u = a cos φ, v = −a sin φ,

.

(4.17)

where .ω is the natural frequency corresponding to the linearized master mode. The constraint equations are written in terms of the master pair, formed by .a and .φ as: x = Pi (a(t), φ(t)),

. i

yi = Q i (a(t), φ(t)), i = 1, 2, . . . , n.

(4.18)

The approximation of the constraint equations is given by the Galerkin method, using a double series in terms of the amplitudes and modal phase angles: n

.

Pi (a(t), φ(t)) =

na φ ∑ ∑

Cilm Tl,m (a(t), φ(t)), i = 1, 2, . . . , n,

(4.19)

Dilm Ul,m (a(t), φ(t)), i = 1, 2, . . . , n,

(4.20)

l=1 m=1 na nφ

Q i (a(t), φ(t)) =

∑∑

l=1 m=1

where .Cilm and . Dilm are coefficients of the expansions to be determined and .Tl,m and .Ul,m are the shape functions of the Galerkin method, typically composed of harmonics functions in terms of .a and .φ, defined in general in a pre-chosen domain .a ∈ [0, a0 ], φ ∈ [0, 2π]. The parameters .n a and .n denote the number of functions φ in the expansions used for .a and .φ respectively. The substitution of Eqs. (4.19) and (4.20) into the equations that govern the invariant manifold results in equations in terms of the modal coordinates that can be solved by numerical integration. The advantage of this method over the asymptotic method is that the convergence domain is known a priori and, therefore, the technique can be used to approach a larger range of motions. However, as the solution is obtained numerically, the advantages of using analytical solutions are lost, mainly for parametric analysis. Jiang [12] generalized the technique based on the Galerkin projection method for multimodal analysis, while Apiwattanalunggarn [1] developed a method simi-

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115

lar to this method based on the Galerkin procedure, using, however, velocity and displacement as modal parameters and not the amplitude and the phase angle. A problem common to many numerical and analytic tools for the analysis of nonlinear dynamical systems is the feasibility of their application in engineering problems possessing a large number of degrees of freedom. Also the difficulties arising from the hyperbolic nature of the manifold-governing PDEs must be taken into account. Procedures to obtain invariant manifolds by the asymptotic method adapted to handle equations of motion of systems discretized by finite element techniques have been proposed by Mazzilli and co-workers [18, 41]. Blank et al. [4] also solved the PDEs in modal space numerically. They write the PDEs in terms of transport equations and discretize those using finite differences. Renson et al. [34] propose a finite-element-based algorithm, combining the streamline upwind Petrov– Galerkin method with mesh moving and domain prediction–correction techniques to solve the manifold-governing partial differential equations, while Kuether and Allen [15] present a numerical approach to directly compute nonlinear normal modes of geometrically nonlinear finite element models.

4.7 Multi-mode—Asymptotic Approach One important feature of nonlinear continuous structural systems is the presence of internal resonances, which may occur if the linear natural frequencies are commensurate or nearly commensurate. Of particular importance, due to its influence on the structural response, are the .1:1 internal resonance, usually associated with system symmetries; the .1:2 internal resonance, caused be quadratic nonlinearities; and the .1:3 resonance arising from cubic nonlinearities [24, 25]. Multimodes are a result of the search for maintaining the invariance property in the face of the presence of internal resonances [7, 8, 29], and multimodal analysis is, in general, used to remove the contamination problems between non-simulated modes and simulated ones [5]. The construction of the multimodal invariant manifolds is obtained by choosing the . Mm modes of interest, simulated modes or master modes, which are brought together in a subset . S M . The master pairs related to these modes are expressed by the vectors .{u M } and .{v M }, so that the subspace of the invariant manifolds associated with the multimode has a dimension of .2Mm within the phase space of the .mdof system. The procedure for obtaining the invariant manifolds is similar to that presented in the previous section, but now the parameterization of the invariant manifolds is obtained by using . Mm displacement-velocity pairs: x = ui ,

. i

yi = vi ∀i ∈ S M .

(4.21)

In addition, slave coordinates are parameterized by the following constraint functions: . x i = Pi ({u M }, {v M }), yi = Q i ({u M }, {v M })∀i ∈ / SM . (4.22)

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As in the case of individual modes, by replacing Eqs. (4.21) and (4.22) in the equation of motion of the system, and using the chain rule to determine the variables, the following partial differential equations governing the invariant manifolds are obtained: .

Qi =

∑ ∂P ∂ Pi i vk + f , ∂u k ∂vk k k∈S

(4.23)

∑ ∂Q ∂ Qi i vk + f . ∂u k ∂vk k k∈S

(4.24)

M

fi =

M

Expansions in Taylor series are used in terms of the master coordinates. The number of coefficients in the expansions grows considerably, depending on the number of modes included. As the multimode is reduced to a singular normal mode when only one mode is included in the analysis, an alternative is to obtain the invariant manifolds individually and use the coefficients of their restriction functions to obtain the multimodes, thus reducing the computational effort for the system solution [5]. According to Boivin et al. [5], non-simulated modes can be seen in two ways: as a set, i.e., as another multimode, or separately as individual normal modes. In this way, normal modes or individual invariant manifolds can be considered as a particular case of multimodes. There are two extreme cases: the first where all modes are simulated, so that the complete system is analyzed and represents the largest possible multimode for the problem, and the second in which only one mode is simulated, obtaining a system with a degree of freedom. As in the singular modal analysis, multimodal analysis represents a local solution and its precision depends both on the amplitude of the motion and on the proximity of internal resonances between the simulated and non-simulated modes [31]. Application of nonlinear vibration modes to conceptual models of offshore structures applying the asymptotic and Galerkin method can be found in [7, 8].

4.8 Computation of NNMs Using Numerical Continuation Techniques The frequency-energy plots (FEP) have been used to study the evolution of the NNMs as a function of the total energy of the system (function of the initial conditions) and to identify possible bifurcations of these modes, which is common at high energy levels. They can be obtained by continuation algorithms [30]. Here a modified algorithm for continuation of periodic solutions of conservative systems is described. The algorithm involves applying the shooting method with a continuation algorithm consisting of a series of prediction and correction steps to obtain all periodic solutions for increasing levels energy levels, each defined by a set of initial conditions, . x 0 . Each periodic solution corresponds to a closed orbit in phase space with period

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T . Unlike forced motion, the period .T of the free response is not known a priori. The shooting method consists in finding, in an iterative way, the initial conditions .x0 and the period .T that satisfy the two-point boundary-value problem defined by the periodicity condition . H (x 0 , T ) = ϕ(x 0 , T ) − x 0 = 0. (4.25)

.

The method relies on numerical time integration of the equation of motion together with the Newton–Raphson algorithm. In general, the initial guesses .x0 and .T do not satisfy the periodicity condition, and a residual of the analytic function. H is obtained. The initial conditions can be chosen anywhere along the periodic solution, thus the phase is arbitrary. Hence, an additional condition must be specified in order to remove the arbitrariness of the initial conditions. Therefore, a phase constraint .h(x) is defined to obtain the solution belonging to the orbit. Thus, the calculation of nonlinear modes consists of continuously solving the boundary value problem with the following constraints { .

F(x0 , T ) =

H (x0 , T ) h(x0 ) = 0

.

(4.26)

The correct definition of the phase constraint is essential for the analysis. In the considered examples, the phase condition .h(x) = is satisfied imposing zero initial velocities. The predictor step consists of performing an Euler step of specified value. It starts with a known, very low-energy solution. The direction of continuation is given by the solution of { ∂x H (x0 , T0 )Δx + ∂t H (x0 , T0 )ΔT = 0 , |||| = 1. (4.27) . ∂x h(x0 )Δx = 0 In order to fix the solution vector, the predictor tangent [23] is adopted. Therefore, the system is transformed into { .

∂x H (x0 , T0 )z = −∂t H (x0 , T0 )ΔT ∂x h(x0 )z = 0

, ΔT = (1 + z·)− 2 , Δx = zΔT, (4.28) 1

ensuring that the vector . is unitary and tangent to the solution curve in the expanded space. The next solution is then estimated as x (0) = x0 + ΔxΔT,

(4.29)

T1(0)

(4.30)

. 1

= T0 + ΔT Δs.

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The correction step consists of decreasing the residue given by . F(x1 , T1 ), since the points obtained in the predictor step are approximations. The correction vector is given by { .

∂x H (x1(i) , T1(i) )dΔx + ∂t H (x1(i) , T1(i) )dΔT = −H (x1(i) , T1(i) ), ∂x h(x1(i) )dΔx = −h(x1(i) ), Δx · dΔx + ΔT · dΔT = −g(x1(i) , T1(i) ),

, (4.31)

where the residual function is given by [23] g(x1(i) , T1(i) ) = (x1(i) − x0 )Δx + (T1(i) − T0 )ΔT − Δs.

.

(4.32)

This expression defines a continuation by pseudo-arclength. In [30], the function is taken as .g = 0, which restricts possible solutions to the orthogonal plane to the tangent vector . at the point estimated in the predictor step, .. Differently, the present strategy allows solutions outside the orthogonal plane to be obtained, as long as they are within a hypersphere of radius .Δs. This strategy proved to be numerically more stable, mainly in problems where the internal resonance implies a multiplicity of nonlinear modes. The estimated solution is then corrected by x (i+1) = xi(i) + dΔx,

. 1

T1(i+1)

=

Ti(i)

+ dΔT.

(4.33) (4.34)

The correction phase is repeated until the following condition is satisfied || ( )|| || (i) (i) || || H x1 , T1 || || || ≤ ε, . || (i) || || x 1 ||

(4.35)

where .ε is a sufficiently small number. In many cases, the smallness of the residual means that the approximation is sufficiently close to the true solution. Adaptive step size and directional controls are adopted in order to maintain a favorable ratio between computational cost and accuracy of the results [30].

4.9 Illustrative Examples Most examples of the NNMs in the technical literature deals with low-order lumpedmass systems (typically with a few degrees of freedom) or focuses on specific resonances of distributed parameter systems. Two 2dof models analyzed in several papers [13, 21, 22, 28, 42] are illustrated in Fig. 4.2. The model in Fig. 4.2 consists in two unitary masses, three linear springs and an additional spring with cubic nonlinearity.

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119

Fig. 4.2 Asymmetric nonlinear 2dof system with cubic nonlinearity

This localized nonlinearity is sufficient to modify drastically the dynamics of the system. The potential energy, .U , and kinetic energy, .T , are given by: U=

.

1 2 1 1 1 1 1 x + (x − x2 )2 + x22 + x14 ; T = x˙12 + x˙22 . 2 1 2 1 2 8 2 2

(4.36)

Based on the Lagrange function . L = T − U and applying Hamilton’s principle the following equations of motion are obtained: x¨ + (2x1 − x2 ) + 0.5x13 = 0, x¨2 + (2x2 − x1 ) = 0.

. 1

(4.37) (4.38)

The conservation of energy principle leads to: U +T =

.

1 1 1 1 1 2 1 x1 + (x1 − x2 )2 + x22 + x14 + x˙12 + x˙22 = h = const. (4.39) 2 2 2 8 2 2

The linearized system has the following two linear frequencies and vibration modes: rad = 0.16 Hz; ϕ1T = α{1 1}, s √ rad ω2 = 3 = 0.27 Hz; ϕ2T = β{1 − 1}. s

ω1 = 1

.

(4.40) (4.41)

Figure 4.3 illustrates the two vibration modes; in the first one the motion is inphase and in the second one the motion is out-of-phase. In this case there is no similar mode. There are two non-similar nonlinear modes, which are the continuation of the two linear modes. For the nonlinear system, the modal lines are curves, resulting from the nonlinear relationship between the coordinates during the periodic motion. These curved NNMs, termed non-similar NNMs by Rosenberg, are generic in nonlinear systems, and their existence complicates the concept of orthogonality between modes. The two Poincaré sections in Fig. 4.4 show the existence of two stable similar modes separated by saddles. Considering the following approximation for the two degrees of freedom

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P. B. Gonçalves

(a) in-phase (x1 , x˙ 1 , x2 , x˙ 2 ) = (1, 0, 1, 0)

(b) modal line

(c) out-of-phase (x1 , x˙ 1 , x2 , x˙ 2 ) = (1, 0, −1, 0)

(d) modal line

Fig. 4.3 Linear vibration modes

(a) Fig. 4.4 Poincaré sections of the 2dof nonlinear system

(b)

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Fig. 4.5 Nonlinear frequency-amplitude relation–hardening nonlinearity

x = A cos(ωt), x2 = B cos(ωt) = 0,

. 1

(4.42)

and applying the harmonic balance method, the following modal amplitudes are obtained / A 8(ω 2 − 3)(ω 2 − 1) , B= . (4.43) .A = ± 2 3(ω − 2) (2 − ω 2 ) Based on these expressions, Fig. 4.5 shows the nonlinear frequency-amplitude relation which are favorably compared with the results obtained by the shooting method up to large amplitude vibrations. Both modes show a hardening behavior in agreement with the positive cubic nonlinearity. The frequency-energy plot (FEP) is a useful tool in analyzing nonlinear system response. Figure 4.6a shows the variation of the two vibration frequencies with the energy level, obtained using the continuation algorithm presented in Sect. 4.8. Due to the nonlinear variation of the two frequencies, . p:q internal resonance may occur for higher energy levels, even though the linear natural frequencies are not commensurate. Figure 4.6b shows in detail the FEP associated with the in-phase linear mode. It is observed that at high energy levels internally resonant NNMs are observed (U21, S31, U41, S51). A detailed analysis of these resonant cases can be found in [14]. Figure 4.7 shows the evolution of the nonlinear vibration mode associated with the first linear mode with increasing energy levels, displaying the increasing complexity of the nonlinear free vibration response. The coordinates used to obtain the NNMs are those calculated by the continuation algorithm in Sect. 4.8. Both the time response

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(b) Detail of the in-phase mode (ω0 = 1)

(a) Frequency-energy plot – 2 NNMs

Fig. 4.6 Variation of the nonlinear vibration frequency with the energy level

of the two coordinates (first column–.x1 in red and .x2 in blue) and the configuration space are shown (second column). A particular feature of these NNMs is that their shape depends on the total energy of the system, .h. When special spatial symmetries exist, the NNMs may degenerate into (energy-invariant) straight modal lines, as in the linear case. Now the derivation of the two-dimensional invariant manifolds using the asymptotic method described in Sect. 4.5 are obtained. Considering the pair of master coordinates .x1 = u, x˙1 = v as control variables, the two nonlinear modes expanded up to the .5th order are: Mode 1: u˙ = v,

(4.44)

.

v˙ = −u −

3

2

5

4

u uv uv u 3 + + − u 3v2 + , 3 4 66 44 352

(4.45)

with controlled variables given by x = u,

(4.46)

. 1

x˙1 = v,

(4.47) 3

2

5

4

u uv uv u 3 + + − u 3v2 + , 3 4 66 44 352 v3 u4v u 2 v3 v5 x˙2 = v + + − + . 4 22 11 352

x2 = u +

(4.48) (4.49)

Mode 2: u˙ = v,

.

(4.50)

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v˙ = −3u − 0.3076923077u 3 + 0.05769230769uv 2 − 0.01967055813u 5

.

− 0.01727170958u 3 v 2 − 0.002113985287uv 4 ,

(4.51)

and

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 4.7 Evolution of the in-phase NNM with the energy level. S–stable, U– unstable. a S1. IC: .(x1 , x˙1 , x2 , x˙2 ) = (0, −2.4065, 0, −12.7812), 1.3105 Hz, b S2. IC: .(x 1 , x˙ 1 , x 2 , x˙ 2 ) = (−0.0005, 16.4534, 0.0025, −79.2301), 1.4071 Hz, c S3. IC: .(x1 , x˙1 , x2 , x˙2 ) = (0, 3.0357, −0.0001, −273.9883), 1.4018 Hz, d S4. IC: .(x 1 , x˙ 1 , x 2 , x˙ 2 ) = (−0.3591, 64.3154, −0.0022, −148.6566), 1.4117 Hz, e S5. IC: .(x 1 , x˙ 1 , x 2 , x˙ 2 ) = (0, 11.2889, −0.003, −866.3097), 1.4084 Hz, f U1. IC: .(x 1 , x˙ 1 , x 2 , x˙ 2 ) = (3.6273, 4.1972, 0.0404, −7.2279), 1.3943 Hz

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(g)

(h)

(i)

(j)

(k)

(l)

Fig. 4.7 (continued)

. 1

x = u,

(4.52)

x˙1 = v,

(4.53)

x = −u + 0.1923076923u 3 + 0.05769230769uv 2 − 0.01967055813u 5

. 2

− 0.01727170958u 3 v 2 − 0.002113985287uv 4 ,

(4.54)

x˙ = −v + 0.2307692308u 2 v + 0.05769230769v 3 − 0.03022549176u 4 v

. 2

− 0.01979050056u 2 v 3 − 0.002113985287v 5 .

(4.55)

4 Nonlinear Normal Modes and Reduced Order Models

(a)

125

(b)

Fig. 4.8 Two-dimensional invariant manifolds of system with the planes corresponding to the LNMs Fig. 4.9 Symmetric nonlinear 2dof system

The two-dimensional invariant manifolds of system are illustrated in Fig. 4.8 together with the planes corresponding to the LNMs. A fundamental property of NNMs is that their number may exceed the number of DOFs of the system. So, not all NNMs can be regarded as nonlinear continuation of normal modes of the associated linear systems, due to mode bifurcations or internal resonance. In many cases this is connected to symmetries of the mechanical system [28]. Consider the discrete system in Fig. 4.9, with three nonlinear cubic springs given by . f i (u) = f 1i u + f 3i u 3 . If the linear part of the connecting spring is different from zero, . f 12 /= 0, the linearized problem has two vibration modes and the associated nonlinear problem two NNMs which are continuation of the linear modes, as in the previous example. However if the connecting spring is purely nonlinear,. f 12 = 0, (notice that in this case the linearized system is composed of two independent linear spring-mass systems), the number of nonlinear modes can be higher than the number of degrees of freedom, depending on the stiffness of the nonlinear spring. Consider that . f 32 = K , the total energy of the system in then given by U +T =

.

1 2 1 4 K 1 1 1 1 x + x + (x1 − x2 )4 + x22 + x24 + x˙12 + x˙22 = h = const. 2 1 4 1 4 2 4 2 2 (4.56)

and the associated nonlinear equations of motion are

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x¨ + x1 + x13 + K (x1 − x2 )3 = 0,

(4.57)

x¨2 + x2 +

(4.58)

. 1

x23

+ K (x2 − x1 ) = 0. 3

It is observed through the mass and stiffness matrices of the linearized system that the two equations are decoupled and the coefficients are identical for the two equations: [

] [ ] 10 10 .M = ,K = , 01 01 |K − λM| = (λ − 1)2 , λ1 = λ 2 =

ω12

=

ω22

= 1.

(4.59) (4.60) (4.61)

Thus a degenerate eigenvalue problem is obtained where the eigenvalues are repeated, i.e. the system has both natural frequencies identical, leading to .1:1 internal resonance. In such case, any vector is an eigenvector. This system possesses similar NNMs that obey to the relation x = cx1 .

(4.62)

. 2

Substituting the linear relation into (4.62) and (4.57)–(4.58), leads to x¨ + x1 + x13 + K (x1 − x2 )3 = 0,

(4.63)

c x¨1 + cx1 +

(4.64)

. 1

cx13

+ K (cx1 − x2 ) = 0. 3

Because both equations must lead to the same solution, it follows that 1 + K (1 − c)3 x13 =

.

K =

−c . (1 − c)2

c3 + K (c − 1)3 3 x1 , c

(4.65) (4.66)

Equations (4.59)–(4.61) means that system (4.57)–(4.58) always possesses two modes characterized by .c = ±1 (in-phase and out-of-phase modes) that are direct extension of the linear modes. However, this system can possess two additional similar NNMs that cannot be captured using linearization procedures. As shown in Fig. 4.10, the in-phase NNM is always stable, while the out-of-phase mode is initially unstable up to . K = 0.25, after which value it becomes stable. At . K = 0.25 there is a bifurcation where two additional NNMs emanate. For selected values of . K < 0.25 up to four NNMs exist for the 2dof system. For example, considering . K = 0.2 there are two additional modes for the proportionality constants .c = −0.38197 and .c = −2.61803. Figure 4.11 shows for an energy level .h = 0.1 the Poincaré section of the system where three centers (corresponding to the in-phase and the two additional

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Fig. 4.10 The variation of the number and stability of the NNMs with parameter . K

Fig. 4.11 Poincaré section for . K = 0.2 and .h = 0.1, showing the four similar NNM modes

modes, which are stable) and a saddle (corresponding to the out-of-phase mode, which is unstable) are observed, confirming the results of the previous analysis. Detailed analyses of the two 2dof models illustrated here can be found in several papers. The reader may refer to [13, 21, 22, 28, 42]. In many cases several bifurcations can be observed for increasing energy levels, as observed in Augusti’s model [28], Fig. 4.12a. This 2dof structural system has four planes of symmetry and, under axial load, the pre-buckling configuration loses stability through an unstable bifurcation. At the critical load, due to nonlinear modal coupling, four nonlinear equilibrium paths emerge associated with the four planes of symmetry, Fig. 4.12b. Two unstable uncoupled paths, U1 and U2, and two unstable coupled paths, C1 and C2 [28, 42], Fig. 4.12b. The results for the nonlinear vibration analysis are depicted in Fig. 4.13, which exhibits a behavior closely related to the static case. Figure 4.13a shows the four similar stable modes obtained for low energy

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(a) Augusti’s model

(b) Fundamental (FD) and post-buckling equilibrium paths (U1, U2, C1, C2)

Fig. 4.12 Augusti’s model [28] and nonlinear equilibrium paths of the axially loaded column

levels. Figure 4.13b exhibits the potential energy for a load level lower than the static critical load, exhibiting a minimum associated with the stable fundamental path and four saddles associated with the descending unstable coupled paths. Figure 4.13c–e exhibits the Poincaré sections for increasing energy levels for the similar mode M2 (the same results are obtained for M1). It is observed, as summarized in Fig. 4.12f, that mode M2 undergoes two bifurcations generating an increasing number of nonlinear modes. Other examples of structural systems with inherent symmetries displaying similar behavior can be found in [7, 28]. Other formulations deriving from the Rosenberg and Shaw and Pierre contributions have been proposed in recent years, which can be successfully used to derive consistent reduced order models. The normal form approach [11, 44] has been proposed and applied in model-order reduction and can handle linear modal damping through a change of coordinates. Spectral submanifold (SSMs) introduced by Haller and Ponsioen [9] is another tool and has been effectively applied to dissipative systems. A review of the different methodologies for the derivation of reduced order models based on the nonlinear normal modes concept can be found in the recent comprehensive review by Touzé et al. [45].

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(a) Modal lines

(b) Potential energy

(c) Low energy level

(d) Intermediate energy level

(e) High energy level

(f) Bifurcation

Fig. 4.13 Augusti’s model. a Modal lines of the four similar modes; b potential energy surface and curves of equal potential energy. Si: Saddles. PMi: Stable position corresponding to a local minimum; c–e Poincaré sections of the nonlinear mode M2 for increasing energy level and f bifurcation diagram of the nonlinear mode as a function of the energy level. [28]

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References 1. Apiwattanalunggarn, P.: Model reduction of nonlinear structural systems using nonlinear normal modes and component mode synthesis. Ph.D. thesis, Univesity of Michigan (2003) 2. Apiwattanalunggarn, P., Shaw, S.W., Pierre, C., Jiang, D.: Finite-element-based nonlinear modal reduction of a rotating beam with large-amplitude motion. Journal of Vibration and Control 9(3-4), 235–263 (2003). https://doi.org/10.1177/107754603030751 3. Avramov, K.V., Mikhlin, Y.V.: Review of applications of nonlinear normal modes for vibrating mechanical systems. Applied Mechanics Reviews 65(2) (2013). https://doi.org/10.1115/1. 4023533 4. Blanc, F., Touzé, C., Mercier, J.F., Ege, K., Bonnet-Ben Dhia, A.S.: On the numerical computation of nonlinear normal modes for reduced-order modelling of conservative vibratory systems. Mechanical Systems and Signal Processing 36(2), 520–539 (2013). https://doi.org/10.1016/j. ymssp.2012.10.016. URL https://hal.archives-ouvertes.fr/hal-00772317 5. Boivin, N., Pierre, C., Shaw, S.W.: Nonlinear modal analysis of structural systems using multi-mode invariant manifolds. In: AIAA Dynamics Specialists Conference, Hilton Head, SC, (1994) 6. Boivin, N., Pierre, C., Shaw, S.W.: Nonlinear normal modes, invariance, and modal dynamics approximations of nonlinear systems. Nonlinear dynamics 8(3), 315–346 (1995) 7. Gavassoni, E.: Application of nonlinear vibration modes to conceptual models of offshore structures. Ph.D. thesis, PUC-Rio (2012) 8. Gavassoni, E., Gonçalves, P.B., Roehl, D.M.: Nonlinear vibration modes of an offshore articulated tower. Ocean Engineering 109, 226–242 (2015). https://doi.org/10.1016/j.oceaneng. 2015.08.028 9. Haller, G., Ponsioen, S.: Nonlinear normal modes and spectral submanifolds: existence, uniqueness and use in model reduction. Nonlinear Dynamics 86(3), 1493–1534 (2016). https://doi. org/10.1007/s11071-016-2974-z 10. Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer New York (2013) 11. Jezequel, L., Lamarque, C.: Analysis of non-linear dynamical systems by the normal form theory. Journal of Sound and Vibration 149(3), 429–459 (1991). https://doi.org/10.1016/0022460x(91)90446-q 12. Jiang, D.: Nonlinear modal analysis based on invariant manifolds – application to rotating blade systems. Ph.D. thesis, Univesity of Michigan (2004) 13. Kerschen, G., Peeters, M., Golinval, J., Vakakis, A.: Nonlinear normal modes, part i: A useful framework for the structural dynamicist. Mechanical Systems and Signal Processing 23(1), 170–194 (2009). https://doi.org/10.1016/j.ymssp.2008.04.002 14. Kerschen, G., Peeters, M., Golinval, J.C., Vakakis, A.: Nonlinear normal modes, part i: An attempt to demystify them. In: In: 26th International Modal Analysis Conference, Orlando (2008) 15. Kuether, R.J., Allen, M.S.: A numerical approach to directly compute nonlinear normal modes of geometrically nonlinear finite element models. Mechanical Systems and Signal Processing 46(1), 1–15 (2014). https://doi.org/10.1016/j.ymssp.2013.12.010 16. Lacarbonara, W., Rega, G., Nayfeh, A.: Resonant non-linear normal modes. part i: Analytical treatment for structural one-dimensional systems. International Journal of Nonlinear Mechanics 38(6), 851–872 (2003). https://doi.org/10.1016/s0020-7462(02)00033-1 17. Manevich, L., Mikhlin, I.: On periodic solutions close to rectilinear normal vibration modes. Journal of Applied Mathematics and Mechanics 36(6), 988–994 (1972). https://doi.org/10. 1016/0021-8928(72)90032-9 18. Mazzilli, C.E.N., Neto, O.B.: Evaluation of nonlinear normal modes for finite-element models. Computers & Structures 80(11), 957–965 (2002) 19. Meirovitch, L.: Elements of vibration analysis. McGraw-Hill Science (1975)

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20. Meyrand, L., Sarrouy, E., Cochelin, B., Ricciardi, G.: Nonlinear normal mode continuation through a proper generalized decomposition approach with modal enrichment. Journal of Sound and Vibration 443, 444–459 (2019). https://doi.org/10.1016/j.jsv.2018.11.030 21. Mikhlin, Y.: Nonlinear normal vibration modes and their applications. In: In Proceedings of the 9th Brazilian conference on dynamics Control and their Applications, pp. 151–171 (2010) 22. Month, L.A., Rand, R.H.: An application of the Poincaré map to the stability of nonlinear normal modes. Journal of Applied Mechanics 47(3), 645–651 (1980). https://doi.org/10.1115/ 1.3153747 23. Nayfeh, A., Balachandran, B.: Applied nonlinear dynamics. Wiley, New York (1995) 24. Nayfeh, A.H.: Nonlinear interactions: analytical, computational, and experimental methods. Wiley, New York (2000) 25. Nayfeh, A.H., Balachandran, B.: Modal interactions in dynamical and structural systems. Applied Mechanics Reviews 42(11S), S175–S201 (1989). https://doi.org/10.1115/1.3152389 26. Nayfeh, A.H., Chin, C., Nayfeh, S.A.: On nonlinear normal modes of systems with internal resonance. Journal of Vibration and Acoustics 118(3), 340–345 (1996). https://doi.org/10. 1115/1.2888188 27. Nayfeh, A.H., Nayfeh, S.A.: On nonlinear modes of continuous systems. Journal of Vibration and Acoustics 116(1), 129–136 (1994). https://doi.org/10.1115/1.2930388 28. Orlando, D.: Nonlinear dynamics, instability and control of structural systems with modal interaction. Ph.D. thesis, Pontifícia Universidade Católica do Rio de Janeiro (2010) 29. Orlando, D., Gonçalves, P.B., Rega, G., Lenci, S.: Influence of symmetries and imperfections on the non-linear vibration modes of archetypal structural systems. International Journal of NonLinear Mechanics 49, 175–195 (2013). https://doi.org/10.1016/j.ijnonlinmec.2012.10.004 30. Peeters, M., Viguié, R., Sérandour, G., Kerschen, G., Golinval, J.C.: Nonlinear normal modes, part II: Toward a practical computation using numerical continuation techniques. Mechanical Systems and Signal Processing 23(1), 195–216 (2009). https://doi.org/10.1016/j.ymssp.2008. 04.003 31. Pesheck, E., Boivin, N., Pierre, C., Shaw, S.W.: Nonlinear modal analysis of structural systems using multi-mode invariant manifolds. Nonlinear Dynamics 25(1/3), 183–205 (2001). https:// doi.org/10.1023/a:1012910918498 32. Pesheck, E., Pierre, C., Shaw, S.W.: A new Galerkin-based approach for accurate nonlinear normal modes through invariant manifolds. Journal of Sound and Vibration 249(5), 971–993 (2002). https://doi.org/10.1006/jsvi.2001.3914 33. Rand, R.H.: Nonlinear normal modes in two-degree-of-freedom systems. Journal of Applied Mechanics 38(2), 561–561 (1971). https://doi.org/10.1115/1.3408826 34. Renson, L., Deliége, G., Kerschen, G.: An effective finite-element-based method for the computation of nonlinear normal modes of nonconservative systems. Meccanica 49(8), 1901–1916 (2014). https://doi.org/10.1007/s11012-014-9875-3 35. Rosenberg, R.: On nonlinear vibrations of systems with many degrees of freedom. In: Advances in Applied Mechanics, pp. 155–242. Elsevier (1966). https://doi.org/10.1016/s00652156(08)70008-5 36. Rosenberg, R.M.: Normal modes of nonlinear dual-mode systems. Journal of Applied Mechanics 27, 263–268 (1960) 37. Shaw, S., Pierre, C.: Non-linear normal modes and invariant manifolds. Journal of Sound and Vibration 150(1), 170–173 (1991). https://doi.org/10.1016/0022-460x(91)90412-d 38. Shaw, S., Pierre, C.: Normal modes for non-linear vibratory systems. Journal of Sound and Vibration 164(1), 85–124 (1993). https://doi.org/10.1006/jsvi.1993.1198 39. Shaw, S., Pierre, C.: Normal modes of vibration for non-linear continuous systems. Journal of Sound and Vibration 169(3), 319–347 (1994). https://doi.org/10.1006/jsvi.1994.1021 40. Shaw, S.W., Pierre, C., Pesheck, E.: Modal analysis-based reduced-order models for nonlinear structures : An invariant manifold approach. The Shock and Vibration Digest 31, 3–16 (1999) 41. Soares, M., Mazzilli, C.: Nonlinear normal modes of planar frames discretised by the finite element method. Computers & Structures 77(5), 485–493 (2000). https://doi.org/10.1016/s00457949(99)00233-3

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42. Thompson, J.M.T.: Elastic instability phenomena. John Wiley & Sons, Chichester New York (1984) 43. Touzé, C., Amabili, M.: Nonlinear normal modes for damped geometrically nonlinear systems: application to reduced-order modelling of harmonically forced structures. Journal of Sound and Vibration 298(4-5), 958–981 (2006). https://doi.org/10.1016/j.jsv.2006.06.032 44. Touzé, C., Thomas, O., Chaigne, A.: Hardening/softening behaviour in non-linear oscillations of structural systems using non-linear normal modes. Journal of Sound and Vibration 273(1-2), 77–101 (2004). https://doi.org/10.1016/j.jsv.2003.04.005 45. Touzé, C., Vizzaccaro, A., Thomas, O.: Model order reduction methods for geometrically nonlinear structures: A review of nonlinear techniques. Nonlinear Dynamics 105(2), 1141– 1190 (2021). https://doi.org/10.1007/s11071-021-06693-9 46. Vakakis, A.: Nonlinear normal modes and their applications in vibration theory: An overview. Mechanical Systems and Signal Processing 11(1), 3–22 (1997). https://doi.org/10.1006/mssp. 1996.9999 47. Vakakis, A.F.: Analysis and identification of linear and nonlinear normal modes in vibrating systems. Ph.D. thesis, California Institute of Technology (1991). https://doi.org/10.7907/ 60CA-1Q64 48. Vakakis, A.F., Manevitch, L.I., Mikhlin, Y.V., Pilipchuk, V.N., Zevin, A.A.: Normal modes and localization in nonlinear systems. Wiley, New York (1996) 49. Wiggins, S.: Introduction to applied nonlinear dynamical systems and chaos. Springer, New York (2003)

Chapter 5

An Introduction to Nonlinear System Identification Luis A. Aguirre

Abstract This chapter is an introduction to nonlinear system identification. A somewhat philosophical description of the main five steps to system identification is provided together with references for further reading.

5.1 Introduction System identification is the “art” of building dynamical mathematical models from data, which are measured from a system which, in principle, could be any dynamical system. A typical system identification problem can be divided into five steps: (i) testing and data collection, (ii) choice of model class, (iii) structure selection, (iv) parameter estimation and (v) model validation. One of the aims of this text is to provide some preliminary discussion to each of these steps, with a clear bias towards nonlinear system identification. Black-box and grey-box techniques will be mentioned. In doing this, a somewhat “philosophical framework” will be proposed in order to bring home to the newcomer some of the real challenges of this fascinating field. Such a framework is admittedly subjective, but it is believed that it will prove helpful in understanding a few important problems and fundamental challenges in system identification. The statement of a system identification problem is simple and can be declared thus: given a dynamical system S, possibly nonlinear, from which a set of measured data . Z N is available, find a mathematical model M that represents S in some meaningful way. To build M exclusively from . Z N is a black-box identification problem. In grey-box problems, besides . Z N there will be some other source of information about S. In . Z N , . N indicates the length of the data set and will be omitted in general. L. A. Aguirre (B) Departamento de Engenharia Eletrônica, Universidade Federal de Minas Gerais, Minas Gerais, Brazil e-mail: [email protected]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. R. Castilho Piqueira et al. (eds.), Lectures on Nonlinear Dynamics, Understanding Complex Systems, https://doi.org/10.1007/978-3-031-45101-0_5

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The following points should be noticed: 1. in black-box system identification, because . Z is all there is to build M, all relevant information about S should be in such data. Hence, . Z has to be carefully obtained; 2. the model M must be a member of a model class .Mc that should be consistent with the relevant aspects of S that are aimed at. For instance, if S is dynamical and nonlinear, so must be .Mc ; 3. given a class of models .Mc consistent with S and the modelling aims, there are many specific candidate models, i.e. .{M1 , M2 . . . Mn } ∈ Mc which are not equivalent nor necessarily adequate to represent specific features of S; 4. a parametric model M is composed by variables and parameters which must be chosen and estimated; 5. a mathematical model M is not directly comparable to a physical system S , because they are entities of different natures. How can one decide if M represents S? The five aspects just mentioned are closely related to the five steps of a typical identification problem, listed in the opening paragraph. Each of the aforementioned five steps will be briefly introduced in the reminder of this text. A fuller discussion can be found in [3].

5.2 Testing and Data Collection There are a number of situations in system identification in which one must build the model from historical data. By historical data it is meant data that have been collected not as a result of any specific designed test. For now, we will assume that a specific test can be performed to produce . Z . Before we start, it will be convenient to point out that the data are very often composed by inputs .u and outputs . y. For now, we will restrict ourselves to the SISO (single-input single-output) case, hence N .Z = [u(k) y(k)], k = 1, 2, . . . N . The standard situation in system identification is that both S and therefore M are dynamical. Hence, it is necessary that . Z be produced as a result of a dynamical test on S. That is, the input .u(k) is designed in such a way that the relevant dynamics of S appear in . y(k). A fundamental point here is to realize that because M is built exclusively from . Z (the black-box case), features of S that do not show up in . Z will most likely not appear in M either.

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In discussing this point, it will be convenient to address the linear and nonlinear cases in turn.

5.2.1 Testing Every physical system S is typically nonlinear and time-varying. Let us assume that the intended model M is linear. This means to say that we are interested in the linear aspects of the dynamical behaviour of S. Hence . Z must be consistent with such an aim. In order to guarantee this and avoid that nonlinear features of S appear in . Z , a typical test is to excite S around an operating point, that is to say, that the amplitude range of .u(k) must be limited to a region in which “the nonlinearities of S are not excited”. Also, because M is intended to be dynamical, . Z must have such information. If .u(k) is too slow, the system S will not have any troubles in following the input and the dynamics will not appear in . y(k). When a driver slows down before passing a speed bump the idea is not to excite the dynamics of the vehicle suspension. So, in general, slow input signals result in data with poor dynamical information. On the other extreme the practitioner also faces problems. If the input is too fast, then there is not enough time for the system to react to the input changes and there is no significant energy transfer from the input to the system. In such a situation the data also turn out to be poor from a dynamical point of view. In technical terms: the power spectrum of the input .Φu (ω) = F(ru ), where .ru (τ ) is the autocorrelation function of .u(k), must have sufficient power in the frequency range of interest, that is, in the frequency range of the dominant dynamics of S. If .Φu (ω) is nonzero at .n frequencies, then .u(k) is said to be persistently exciting of order n.

In practice, a signal is said to be persistently exciting if it is sufficiently rich in order to facilitate the estimation process. Because M should be time-invariant and S is time-varying, one should guarantee that the time-varying aspect does not appear in . Z . This is typically achieved by performing dynamical tests that are not too long. In other words, the changes in S during the test should be negligible. The newcomer to the theory of dynamical systems should be aware that supposing a time-invariant system does not imply a constant output. What changes if we aim at a nonlinear model? In what concerns the time-varying aspect of S nothing changes because we still aim at a time-invariant model M. Hence

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the period during which the data . Z are collected still needs to be sufficiently short as to guarantee that it is reasonable to consider that S did not change significantly during that period. Because M is now nonlinear, then it is required that the relevant nonlinear aspects of S be present in . Z . Clearly, the test cannot possibly be performed as for the linear case, on the contrary, a typical test for nonlinear system identification will probably specify large variations in the input signal .u(k) in order to excite the nonlinearities of S and guarantee that they appear in the data . Z . From a practical point of view, it is often difficult and unsafe to drive the system S over a wide range of operating conditions. This is one of the practical challenges faced by the practitioner that aims at building nonlinear models from data. Fortunately, there are possibilities that help to face such challenges. A common one is to perform several low amplitude tests over a set of operating points that cover the region of interest. The inconvenience of this is that the test can turn out to be long. Another solution to this problem based on grey-box techniques [3]. If from an amplitude point of view the testing of nonlinear systems is more challenging than for linear systems, in terms of frequency content of the input, nonlinear systems are somewhat easier. The reason for this is that nonlinear systems transfer spectral power among different frequencies, which does not happen in linear systems. To see this, suppose we choose as input .u(k) = A cos(ω0 t) with a sufficiently small value of . A in order not to excite the nonlinearities in S. Linear systems theory tells us that in steady-state the output will be of the form . y(k) = a0 cos(ω0 t + φ0 ). In other words, the gain at frequency .ω0 is .a0 /A and the phase at that frequency is .φ0 (usually a negative value). That is we have identified the Bode diagram (frequency response) of S only at .ω0 . Now it should become clear why .u(k) should be persistently exciting of large order .n: to have information at .n different frequencies. If . A is increased nonlinearities will be excited. For linear models input amplitudes should be low and the spectrum should be wide; for nonlinear models the amplitude profile should be large whereas the input spectrum can sometimes be narrow. As a final remark in this section, what if we only have historical data in hand? The same principles apply, but now instead of performing a test to measure . Z with the desired features, one must go through all the available data and find windows that present the desired characteristics. Such windows are candidates to compose . Z . Procedures for detecting transients from a historical record can be found in [32, 74].

5.2.2 Choosing the Sampling Period When it comes to choosing the sampling time .Ts , one immediately thinks in terms of Shannon’s sampling theorem: a signal that does not have any components of frequency above . f max = 1/2Ts can be unambiguously reconstructed from a set of

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samples regularly spaced in time by.Ts . In many practical problems, including system identification, this result is not totally practical for a couple of reasons. First and foremost, . f max is generally not known beforehand, second, to sample a signal with a frequency just above .2 f max is a lower bound rather than a comfortable working value. On the other hand someone might suggest oversampling the data. This also has its troubles as consecutive samples are highly redundant and therefore the numerical problems that must be solved typically become ill-conditioned. A practical procedure that works well in many situations is the following. First, using an admittedly short sampling time record an oversampled data set . Z ∗ = [u ∗ (k) y ∗ (k)], k = 1, 2, . . .. Now we want to choose a decimation factor .Δ ∈ N such that .u(k) = u ∗ (Δk) and ∗ . y(k) = y (Δk). It is assumed that all signals are band-limited. First, the following covariance functions are computed | | r (τ ) = E (y ∗ (k) − y ∗ (k))(y ∗ (k − τ ) − y ∗ (k)) | | r y ∗2' (τ ) = E (y ∗2 (k) − y ∗2 (k))(y ∗2 (k − τ ) − y ∗2 (k)) , . y∗

(5.1)

where E[.·] indicates the mathematical expectation and the overbar indicates timeaverage. The first expression in (5.1) is the standard linear covariance function, whereas the second is a nonlinear function. Notice that such functions are computed using the oversampled data. Second, plot functions .r y ∗ (τ ) and .r y ∗2' (τ ). Call .τ y ∗ .τ y ∗2' the lags at which the first minimum of each function occurs. Choose the least of them, that is .τm∗ = min[τ y ∗ , τ y ∗2' ]. Call .τm the corresponding value for the data decimated with factor .Δ. Hence, we wish to determine .Δ such that 10 ≤ τm ≤ 20,

.

(5.2)

where the limits can sometimes be relaxed to 5 and 25. For example, consider an oversampled signal . y ∗ (k), suppose that the smallest lag corresponding to the first minimum is .τm∗ = 55, hence if we choose .Δ = 4, this value for the decimated signal will be .τm ≈ 14, which satisfies (5.2). Hence the decimation factor could be.Δ = 4. The decimation factor applied must be the same for all signals. Some technical aspects of testing can be found in [45, 54] and specifically for closed-loop systems in [33]. The decimation criterion discussed in this section was originally put forward in [2] and some effects on aspects of system identification have been discussed in [24].

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5.3 Choice of Model Class This is probably the step which receives less attention and it is not really difficult to see why. From a more technical point of view, it suffices that the model class .Mc be sufficiently general to include the relevant aspects of S. From a practical point of view, the practitioner tends to use the model class that he/she is familiar with. Can we be more specific and “scientific” in choosing the model class? Probably the most important choice is to decide if .Mc will be linear or not. This of course has to do with the intended use of the model. For instance, although linear models are admittedly less effective in representing systems in general, they could be preferred, say, for control purposes. Within the class of linear models, there are some subclasses as, for instance, transfer functions and state space representations. If a model is intended to implement a Kalman filter, then .Mc is likely to be the class of linear models in state-space form [35, 78]. In the realm of nonlinear model classes, the variety is very large, in what follows only a few are mentioned: Volterra series [37], nonlinear output frequency response functions [21], Hammerstein and Wiener models [9, 16]. These model classes have been surveyed in [22]. Polynomial and rational NARMAX (nonlinear autoregressive moving average models with exogenous variables) [25, 29, 53]. A recent overview of such model classes and related methods can be found in [23]. Differential equations (continuous-time models) have been used in [47, 63], radial-basis functions [39, 69] and neural networks [17, 26, 40, 67]. The main motivation for using such nonlinear representations is that they are universal approximations, hence regardless of the features of S, such model classes are sufficiently general to represent the system. The model classes mentioned in the previous paragraph can be classified as global in the sense that a single model structure is used to map the whole set of inputs to the output. On the other extreme, one finds models that are a combination of linear models that are responsible for representing the data only locally [19, 20, 61]. See [46, 52] for some early works on this topic. An intermediary class is that of regional models [76]. The model class can be chosen based on the ease with which auxiliary information can be used in the process of model building [3]. In order to facilitate discussions about structure selection and parameter estimation, it will be convenient to choose a working model class. Although most of what will be said is applicable to any model class, in what follows we shall consider the class of NARMAX models [38] .

y(k) = F l [y(k − 1), . . . , y(k − n y ), u(k − d). . . u(k − n u ), e(k − 1), . . . e(k − n e ), e(k)],

(5.3)

where .e(k) accounts for uncertainties and possible noise. Also .n y , n u and .n e are the maximum lags in each variable, and .d ≥ 1 is the pure delay. The function . F can be any nonlinear function, such as a neural network, radial basis function, rational or polynomial, each of which determines a different model class. For the sake of discussion, in what follows . F l [·] will be a polynomial with nonlinearity degree .l.

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5.4 Structure Selection Having decided which model class to use, it is important to realize that there are scores of different model structures within the chosen class.Mc , that is.{M1 , M2 . . . Mn } ∈ Mc , where .n can be very large. On its own that remark justifies the search for a much smaller set of model structures that should be considered in a given situation. However the problem is indeed far more critical than what it seems at first sight. If the model class .Mc is very general, the temptation to grow the model more than needed is great. This is known in the literature as overparametrization or overfitting and often has severe detrimental effects on the dynamics of the identified models [7]. In this section we will provide a brief discussion for model classes that are linear-inthe-parameters to give the reader a feel for the problem. In the next section it will be explained why it is relatively “natural” to overparametrize a model. We start by assuming that .Mc is the class of single-output NARX polynomials. Hence, the corresponding model–see (5.3)–can be rewritten as: .

y(k) = θ1 ψ1 + θ2 ψ2 + . . . + θn θ ψn θ + e(k)

= [ψ1 ψ2 . . . ψn θ ]θ + e(k) = ψ(k − 1)T θ + e(k),

(5.4)

where .ψi is the .ith regressor and .θi the corresponding parameter which can be written in vector form as .θ = [θ1 θ2 . . . θn θ ]T . The (column) vector of regressors is .ψ(k − 1), indicating that all regressors are taken at most up to instant .k − 1. The regressors are any combinations up to degree .l of input and/or output variables down to lags .n u and .n y , respectively. For instance, .ψ1 = y(k − 1), .ψ2 = u(k − 3) and 2 .ψ3 = y(k − 1)u(k − 2) are possible regressors for .n y = n u = l = 3. Hence, the regressors of model (5.4) may contain any combination of lagged inputs, outputs and noise terms up to degree .l. The number of such combinations is determined by the values of .l, .n y , n u and .n e and can easily include thousands of candidate regressors. This huge amount of terms is a major impediment to the usefulness of the estimated model and some kind of mechanism is called for in order to automatically choose the best .n θ regressors to compose the model. This problem is often referred to as model structure selection and must be judiciously accomplished regardless of the mathematical representation being used. This model class is said to be linear-in-the-parameters because all the known parameters can be separated from the known regressors, as seen in (5.4). In the case of model classes that are not linear in the parameters, typically we would find unknown parameters in .ψ(k − 1). The fact that a model be linear-in-the-parameters does not mean that it satisfies the superposition principle, on the contrary. Many linear-in-the-parameters model classes are strongly nonlinear.

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5.4.1 The ERR, SRR and SSMR Criteria Because of their usefulness and wide acceptance, three criteria for choosing the regressors of NARX polynomial models are briefly described next. A widely used criterion in the structure selection is the error reduction ratio (ERR) [27]. This criterion, which is based on one-step-ahead prediction error minimization, evaluates the importance of the model terms according to their ability to explain the output variance. The reduction in the variance of the residuals, that occurs as new terms are included in the model, can be normalized in relation to the output variance .σ 2y . Then, the error reduction ratio due to the inclusion of the .ith regressor in the model can be written as: MS1PE(Mi−1 ) − MS1PE(Mi ) , i = 1, 2, . . . , n, (5.5) .ERRi = σ 2y where .MS1PE(Mi ) stands for the mean square one-step-ahead (OSA) prediction error of the model with .i terms (regressors); .n is the number of candidate terms tested for; and .Mi represents a family of models with nested structures, thus .Mi−1 ⊂ Mi . In (5.5) the numerator equals the reduction in variance of the residuals due to the inclusion of the .ith regressor. A somewhat related criterion, called simulation error reduction ratio (SRR), was defined in [71] as: SRRi =

.

MSSE(Mi−1 ) − MSSE(Mi ) , i = 1, 2, . . . , n, σ 2y

(5.6)

where .MSSE(Mi ) stands for the mean square simulation error of the model with .i terms (regressors). In (5.6) the free-run simulation is used. In Sect. 5.5 the important difference between OSA prediction and simulation errors will be pointed out. In the same vein, the simulation similarity maximization rate was proposed as [14]: SSMRi =

.

Vˆσ (y, yˆi+1 ) − Vˆσ (y, yˆi ) , σ 2y

i = 1, 2, . . . , n,

(5.7)

where . yˆi is the free-run simulated output of the current model and . yˆi+1 refers to the model that has the same regressors as the one that produced . yˆi with the addition of the regressor that is being tested and .n is the number of candidate regressors. In (5.7) ˆσ (X, W ) is the correntropy between the random variables . X and .W and quantifies .V the average similarity between them. The SRR is effective in non ideal identification conditions and often yields more compact models. On the other hand, such a criterion requires a significantly greater computational effort. To partially circumvent such a limitation alternative procedures have been proposed [34, 43]. The SSMR, as the SRR, benefits from using free-run

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simulated model outputs. Also, it is less sensitive to eventual problems caused by non-Gaussianity which is the norm in the nonlinear context. See [13, 48, 51, 64, 66, 70, 79, 81] for a comparison of methods and some recent techniques on structure selection approaches somewhat related to the ERR, SRR and SSMR methods.

5.4.2 Other Criteria Twin concepts that were developed to aid in structure selection problems are the term clusters, indicated by .Ω y p u m , and the respective cluster coefficients, by .Σ y p u m . Terms of the form . y p (k − τ j )u m (k − τi ) ∈ Ω y p u m for .m + p ≤ l, where .τi and .τ j are any time lags. For instance, for the model . y(k) = θ1 y(k − 1)y(k − 2) + θ2 y(k − 1)u(k − 2) + θ3 y(k − 3)u(k − 3) we have .n y = n u = 3, d = 2, l = 2. This model has two term clusters, namely:.Ω y 2 with coefficient.Σ y 2 = θ1 and.Ωuy with coefficient .Σuy = θ2 + θ3 . Often the coefficients of spurious term clusters become very small or oscillate around zero as the model increased in size, hence this could be used to aid in detecting the order of linear models [1] or to detect and discard term clusters in nonlinear system identification [8, 11]. Later on, these concepts turned out to be very useful in grey-box identification problems, as will be discussed later. The use of correlation test has been discussed in [55, 77], but the procedure is less specific than what one would like. In semiphysical modeling prior knowledge of the system is used to establish suitable, though usually nonlinear, terms of the measurements in order to improve on the model structure [60]. In other words, in semiphysical modeling physical insight of the system being modeled is used to determine key term clusters. Of course, semiphysical modeling falls into the category of grey-box techniques. Following a probabilistic framework, the Randomized algorithm for Model Structure Selection (RaMSS) was proposed recently [42]. The method was formulated for the NARX polynomial model class and extended to cope with NARMAX polynomial model class in [73]. The bias/variance dilemma, originally discussed in the context of neural networks [44], is useful for understanding the pros and cons of slight overparametrization. Such a dilemma underlines several so called information criteria such as Akaike’s criterion [12]. By explicitly including the number of model terms, thus increasing the learning dimension, Hafiz and co-workers have used particle swarm algorithms to address the problem of structure selection in nonlinear system identification [49]. The problem of determining the model structure clearly extends to other model classes. For instance, in the case of neural networks there is an extensive literature in pruning methods [72]. The ERR has been adapted for choosing the centres of radial basis function networks [4, 39]. Auxiliary information regarding fixed points

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and steady-state at large provides important clues for structure selection in the case of polynomial models [3]. In closing this section it is important to point out that for a set of data of limited size and quality there is more often than not more than one model structure that is compatible with the data [18], also see examples in [15, 42]. Hence there is uncertainty not only on the model parameters, but also on the model structure. To quantify such uncertainty remains an open problem [18, 48].

5.5 Parameter Estimation By now we have chosen a model class .Mc and a specific model structure within that class. Of course, in practice we usually choose more than a model structure to work with, but for the sake of argument, let us focus on a single model structure. As seen in (5.4) the model is composed by a structure (regressor variables .ψi ) and parameters that are grouped in a vector .θ. Hence our model can be represented as .M(θ) and now the last aspect of the model that still needs determining is .θ. In what follows some of the challenges that must be faced in estimating .θ will be discussed before mentioning some classical estimators.

5.5.1 Underlying Issues As in Sect. 5.1, we start with a philosophical discussion which hopefully will prove helpful. In a very loose and intuitive way, we can say that both the structure of M and ˆ be a good its parameters .θ should be chosen such that the estimated model .M(θ) representation of the system S. The hat on .θˆ indicates that it is an estimated value of .θ. Since we assume that at this stage the model structure has been chosen, we move on to estimate the parameters. Hence we could imagine a naïve procedure by which ˆ ≡ we chose the vector.θˆ that best approximates the model to the system, that is,.M(θ) S. As mentioned before, this is not a viable problem because model and system are of a very different nature. One is a set of equations the other is a set of devices and e.g. physical or biological components, depending on the system. Hence to move towards a solution, we measure from the system S a set of data . Z , but this is still not sufficient because, for the same reason as before, we cannot compare a model ˆ ≡ Z is not a viable problem either. to data, that is, to search for .θˆ such that .M(θ) So, in order to come up with a viable problem, we produce data using the model .Z ˆ –which will be indicated as . Z M for short–and now it is possible to compare M(θ) data with data, because these are of the same nature. Hence, we search for .θˆ such that . Z M is as close to . Z as possible.

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A standard and convenient way of measuring how close a signal is from another is to compute the sum of squared errors:

.

J=

N E |

|2 y(k) − yˆ (k) ,

(5.8)

k=1

ˆ where . y(k) is the measured output and . yˆ (k) is the model output produced with .θ. ˆ Therefore . J indirectly depends on .θ, because . yˆ (k) does. Because . Z = [u(k) y(k)], k = 1, . . . , N and . Z M = [u(k) yˆ (k)], k = 1, . . . , N (notice that .u(k) must be the same in both data sets), for the sake of discussion (5.8) will be represented as . J (Z , Z M ). In this framework, if the data sets are similar then . J (Z , Z M ) is “small”. Or, in other words, the smaller . J (Z , Z M ) the closer the data sets . Z and . Z M are. A remaining key issue is to decide in which way the model will be used to produce the data . Z M . We consider two different ways of doing so by means of a specific and simple example. Consider the model for which the parameters are assumed known .

ˆ yˆ (k) = [y(k − 1) u(k − 1) y(k − 1)u(k − 1)]θ,

(5.9)

hence the right hand side is completely known and, therefore, so is the model output yˆ (k). For example, the model output at time .k = 10 is given by model (5.9) as:

.

.

ˆ yˆ (10) = [y(9) u(9) y(9)u(9)]θ,

where . y(9) is the 9th measured sample of the output, and so on. Because we feed the model with measured data and use it just to move one-step-ahead, the output computed this way is called one step ahead (OSA) prediction and will be indicated by . yˆ1 (k). A different value of. yˆ (10) would be obtained if, using the same model, we compute .

ˆ yˆ (10) = [ yˆ (9) u(9) yˆ (9)u(9)]θ,

where, .

ˆ yˆ (9) = [ yˆ (8) u(8) yˆ (8)u(8)]θ,

and so on. Because this second procedure feeds the model with previously simulated values, the output is often referred to as simulated or free-run output and will be indicated by . yˆs (k). The reader will immediately realize that the ERR criterion is based o . yˆ1 (k) and the SRR and SMRR, on . yˆs (k), see Sect. 5.4.1. What are the main differences between. yˆ1 (k) and. yˆs (k)? Both outputs are produced by the same model and we can think of the data revealing or accumulating “model signatures” or “model features”. Because to produce . yˆ1 (k) the model only has to predict one step into the future starting from measured data, the model features are

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somewhat hard to recognize in . yˆ1 (k). Imagine that the model is unstable, of course the model divergence will be in . yˆ1 (k) when compared to . y(k), but–especially if the sampling time is short–that divergence may not be significant. On the other hand, because . yˆs (k) is produced from previously simulated output, the model features will tend to accumulate in such data which, in a sense, will be far more informative about the model. So, if the model is unstable, . yˆs (k) will diverge to infinity. From the previous discussion there should be no doubts that . yˆs (k) is advantageous over . yˆ1 (k) from a dynamical point of view. Hence it would be nice if we could use the data . Z Ms instead of . Z M1 in estimating the parameter vector. Let us state this more formally in terms of two possible estimators that boil down to two optimization problems: .

θˆ 1 = arg min J (Z , Z M1 ) θ

(5.10)

.

θˆ s = arg min J (Z , Z Ms ). θ

(5.11)

and

For instance, the optimization problem in (5.10) is read thus: the estimated vector θˆ is the argument among all possible .θ that minimizes . J (Z , Z M1 ), and likewise for the problem in (5.11). Since we have already seen that, from a dynamical point of view,. Z Ms is preferable to. Z M1 , it seems natural to conclude that the estimator in (5.11) is probably preferable to the one in (5.10). Well, in fact, the estimator in (5.11) is indeed much more robust to noise and other aspects than (5.10) in many cases [75]. However, the optimization problem in (5.11) is nonconvex, whereas the optimization problem in (5.10) is convex and can be solved in a much simpler way. Hence, to summarize, from a dynamical point of view, the solution to problem (5.11) is preferable, but at the cost of having to solve a nonconvex optimization problem. On the other hand to solve problem (5.10) is numerically preferable, although the solution .θˆ 1 might not be as good and as robust as .θˆ s . Let us close this discussion with three remarks.

. 1

1. Many of the “standard” recommendations in system identification about testing, input persistent excitation, choice of model structure and so on are made in order to improve the chances of.θˆ 1 –which is a numerically inexpensive solution–being such that the model .M(θˆ 1 ) is dynamically acceptable [5]. 2. It must be noticed that for model classes that are not linear-in-the-parameters, even problem (5.10) is a nonconvex optimization problem.

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3. Parameters of chaotic systems can be estimated solving (5.10) because at such short single steps, the effect of the extreme sensitivity to initial conditions is not critical.

5.5.2 Classical Estimators In this section we start by presenting the least squares estimator, which is the classical solution to problem in (5.10), which applies to model classes that are linear-in-theparameters. To see this we start considering the model structure in (5.4) and a set of data . Z N . We can now write (5.4) for any value of .k within the data range. For instance, for .k = 10 we can write . y(10) = ψ(9)T θ + e(10) where .ψ(9) is the vector of regressors which goes up to time .k − 1 = 9 and .e(10) is whatever cannot be explained in the data at time .k = 10 using .ψ(9)T θ. Hence we would like to find .θ in such a way as to minimize the unexplained part .e(10). In order to gain robustness, we consider the model structure (5.4) along all the data, which we now call the identification data set . Z N . That means that for each value of .k within the data range we will have an equation of the form . y(k) = ψ(k − 1)T θ + e(k) which can be expressed as a matrix equation thus: .

y = Ψ θ + e,

(5.12)

N where .Ψ ∈ R N ×n θ , .θ ∈ Rn θ and . y = {y(k)}k=1 . Suppose we have an estimate of the ˆ parameter vector .θ. Then (5.12) can be rewritten as: .

y = Ψ θˆ + ξ = ˆy1 + ξ,

(5.13)

N where .ξ = {ξ(k)}k=1 is the vector of residuals which can be taken as an estimate of the unknown noise .e under “favorable” conditions and . ˆy1 is the vector of OSA predictions. One way of determining .θˆ if .Ψ does not contain columns correlated with .ξ (as in the case of NARX models) is by the Least-Squares (LS) estimator, that minimizes the mean squared value of .ξ = y − ˆy1 and is given by

θˆ

. LS

= (Ψ T Ψ )−1 Ψ T y,

and is the solution to problem (5.10).

(5.14)

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In the classical LS solution (5.14) all the residuals receive equal weight, that is ξ(k) = y(k) − yˆ1 (k), ∀k. There might be reasons in some situations to give specific weights to the residuals as.w(k)ξ(k). Placing.w(k), k = 1, . . . , N along the diagonal of a matrix .W ∈ R N ×N , the solution

.

θˆ

. WLS

= (Ψ T W Ψ )−1 Ψ T W y

(5.15)

is known as the weighted least squares estimator (WLS). There are two classical choices for .W . If the weighting matrix is taken as the covariance matrix for the noise, the WLS becomes the so called Markov estimator which is akin to the generalized least squares estimator. A second important choice of .W is to take the weight of the present moment as.w(k) = 1, the previous one as.w(k − 1) = λ, then.w(k − 2) = λ2 and so on, where .λ < 1 is known as the forgetting factor. Hence the WLS estimator is useful to derive a recursive LS estimator with forgetting factor. The optimization problems in (5.10) and (5.11) are said to be unconstrained. This means that all the degrees of freedom are used to minimize the functional . J , and the solution can be any vector of real values .θˆ ∈ Rn θ . Now suppose that there is a set of constraints on the parameter vector written as .c = Sθ, where .c is a given constant vector, and . S is a known constant matrix. This means that no matter what parameter vector is chosen to minimize . J , it must simultaneously satisfy the set of constraints .c = Sθ. Hence, there are less degrees of freedom available to minimize . J . The solution to the problem .

θˆ CLS =

arg min J (Z , Z M1 ) θ : c = Sθ,

(5.16)

is given by [41] θˆ

. CLS

= (Ψ T Ψ )−1 Ψ T y − (Ψ T Ψ )−1 S T [S(Ψ T Ψ )−1 S T ]−1 (S θˆ LS − c). (5.17)

As it will be shown in future sections, the solution (5.17) turns out to be very helpful when we are able to translate auxiliary information about the system S in terms of a set of equality constraints on the parameters. Of course, this is easier to do for some model classes and hence this serves as motivation for choosing certain model classes depending on the available auxiliary information and our ability to code it in terms of constraints. The issue of parameter estimation–or model training as known in other fields–is vast. A key-point which has not been dealt with because it would require a far more technical discussion has to do with the statistical properties of the noise .e in (5.12). Here it has been implicitly assumed that it is white. Whenever the noise is correlated,

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the LS estimator becomes biased. There are several solutions to this problem, for instance: (i) if one can find a set of variables, called instruments, that are not correlated to the noise and remain correlated to the output, the instrumental variable estimator can be used [83], (ii) the model class can be changed to try to model the correlation in the noise. A common choice is to add a moving average (MA) part to the model. The resulting model class is no longer linear-in-the-parameters and iterative estimators are available such as the extended least squares [38, 62], and (iii) by solving the problem in (5.11) [5]. The basic theory and algorithms can be found in most text-books. There are many alternative procedures for polynomial models such as [68, 80] and also for other representations such as rational [82, 84], neural networks [40, 65] and radial basis functions [39]. We close mentioning that although technical aspects of the estimators certainly depend on the model class used, many discussions in this section, especially those in Sect. 5.5.1 remain valid for other model classes [75].

5.5.3 The Danger of Overparametrization From what has been discussed in this section it is now possible to have a better understanding of the danger of overparametrizing a model. Let us consider problem (5.10) again. Would it be possible to find .θˆ such that . J (Z , Z M1 ) = 0? It might sound absurd (and in practice it is, in fact), but we can force . yˆ1 (k) = y(k), ∀k for the estimation data. Hence, let us see under which conditions.ξ = 0 in (5.13). For this, lets go back one step and consider (5.12). If we increase the size of .θ to the point that N ×N .n = N , then .Ψ ∈ R becomes a square matrix. Assuming that it is nonsingular, θ then .θˆ = Ψ −1 y and therefore .ξ = 0. What we have done is to increase the number of degrees of freedom (parameters to be estimated) to match the number of “constraints” (the number of rows of .Ψ ) to end up with a square system of equations that has a single solution. Is that solution any good? Most likely not. And the reason is that all the uncertainties and noise in the data . y(k) will be fit by the model, which ideally should only fit the underlying dynamics of the system. The resulting model in this hypothetical case has a lot more parameters than needed–remember that.n θ = N –and therefore we say that the model is overparametrized. An interesting remark on which we will not dwell here is this: if we consider problem (5.11) instead, we will not face the same problem. The reason is that in order to have . J (Z , Z Ms ) = 0 we would need to have . yˆs (k) = y(k), ∀k. This will not be achieved in practice because . yˆs (k) is obtained by iterating the model (see Sect. 5.5.1) and an overparametrized model will, more often than not, be unstable, hence . yˆs (k) /= y(k). This discussion then has two messages:

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1. Problem (5.11) is definitely more robust than problem (5.10). 2. If the latter problem is the one being solved, then overparametrization is a real danger.

5.6 Model Validation We finally get to the last step in system identification. Here we would like to answer ˆ represent the underlying dynamics of the system? the question, does the model.M(θ) One of the basic rules in model validation is to use a set of data . Z v for this purpose which is different from the data used to fit the model, that is the estimation or validation data. Z . This is necessary because we cannot compare the model directly to the system S as mentioned before. Then, in order to compare entities that are alike we take data from the system and data from the model . Z M and compare both using some kind of metric .V (Z v , Z M ). It should come as no surprise that the choice of . Z v , . Z M and . V have their influence on the validation. If there is only one set of data . Z N , then this should be divided into a part for identification . Z Ni and a part for N validation . Z v v , such that . N = Ni + Nv . A common choice is to split the data such that . Ni = 2Nv , but this is not a must. As for . Z v , it is recommended that these data be measured from the system operating over a range (amplitude and frequency-wise) consistent with the intended use of the model. In obtaining the identification data . Z it is reasonable to try to excite the system over the widest possible operating range (in the nonlinear case) in order to gather as much information as possible. When it comes to validation, there is no reason why we should request that the model be a faithful representation of the system over operating ranges that will not be visited in practice. Concerning the choice of . Z M , as discussed in Sect. 5.5.1, it should be noted that: the use of . Z Ms should be strongly preferred over . Z M1 . As a matter of fact, the latter is close to useless for model validation, because even poor models can produce OSA predictions that are close to the validation data. On the other hand, the use of. Z Ms is very exacting, and that is what is required in validation. Finally, the use of a chosen metric .V should be considered with care. Suppose that .V is a measure of distance between two data sets. Also, consider two models with . Z Ms1 and . Z Ms2 such that .V (Z v , Z Ms1 ) < V (Z v , Z Ms2 ). This only means that the first model is better than the second with respect to the chosen metric V. A certain metric might not be sufficiently specific to capture the dynamical aspects of the models. Common metrics are the mean squared error or the mean absolute percentage error, but the list is much longer.

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Before mentioning some specific criteria, let it be said that the question: “is this model valid?” should be preceded by the question: “what is the intended use for the model?” In other words, the validity of a model should not be tackled as an absolute attribute, but rather as something that should make sense within a context. That is perhaps one of the greatest difficulties in model validation because very often we do not know exactly what to ask from the model.

5.6.1 Residual Tests This consist of testing the residual vector .ξ for whiteness and for correlations with the input and output [28, 30]. The main idea behind this procedure is that if the model structure is sufficiently flexible and the estimation procedure was sound, then all the correlations in the data should be explained by the model, hence the residuals should not be self-correlated and uncorrelated with the input and output. Also, because the standard linear correlation functions are unable to detect all nonlinear correlations, then specific nonlinear correlation functions should be used [31, 85]. Because the residuals are defined based on the OSA prediction, clearly a model that passes the residual tests might still be inadequate from a dynamical point of view. Also, residual tests are not sensitive to overparametrization. That is, overparametrized models often pass residual tests.

5.6.2 Dynamical Invariants A rather particular situation arises when the system S is chaotic. In that case it is quite common to use a number of dynamical and geometrical invariants to quantify the dynamics. Among such indices one can mention correlation dimension, Lyapunov exponents, Poincaré sections and bifurcation diagrams. Some of theses tools have been discussed in the context of model validation in [6, 50]. A more exacting procedure for model validation, but now restricted to 3D chaotic models is topological analysis [56–58]. This procedure builds topological templates from the system and model data and compares them. From such templates it is possible to extract the population of periodic orbits and to compare the model population

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the that of the original system. This is far more detailed, intricate and complicated than to just compare the appearance of attractors.

5.6.3 Synchronization A very interesting procedure suggested in [36] is that of synchronization. The idea is that that underlies the concept of a state observer. An error signal between model output and data is fed back to the model to force the model output to follow the data. From the days of Huygens it is known that two coupled oscillators will synchronize if their natural frequencies (their intrinsic dynamics) be close. The same applies here. We would like to know if the model dynamics is close to the dynamics underlying the data. Hence we couple model and data and see if the “synchronization error” becomes small. This procedure has two practical problems. 1. In many cases synchronization can be achieved at the expense of increasing the coupling force. It has been pointed out that different dynamics can be made to synchronize and therefore synchronization has little to offer when it comes to (absolute) model validation [10, 59]. 2. If the model does synchronize with large coupling, does this imply that the underlying dynamics have been learned by the model? The answer to this question is negative. However, if the model and data have the same dynamics, the coupling required for synchronization can be very small.

A practical approach to a similar problem has been put forward in [10]. There the main goal is not to validate in absolute terms a model, but rather compare models and choose the one that is closest to the dynamics underlying the data. This was referred to as model evaluation in the aforementioned reference. To achieve this a measure of synchronization effort must be taken into account for the comparison to make any reasonable sense.

Then if the maximum synchronization error is very different, it can be used to rank the models: the best model has the smallest maximum error. However, often models have similar errors, in that case the synchronization effort is used. Hence given two models with similar errors, the one that required a smaller effort is the better model. So, in short synchronization allied to a measure of cost can help in comparing models, but not to get any general assessment of model quality. Acknowledgements Financial support by CNPq and FAPESP (Brazil) is gratefully acknowledged.

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Chapter 6

Asynchronous Modes of Vibration Carlos E. N. Mazzilli, Eduardo A. R. Ribeiro, and Breno A. P. Mendes

Abstract This work provides a comprehensive discussion about the modal asynchronicity (or localisation) phenomenon applied to structural systems. Departing from a Ziegler’s column with two degrees of freedom (DOF)–which nicely illustrates that even simple systems can display asynchronous modes—admissibility conditions for modal asynchronicity are discussed in a series of linear systems, namely a 3DOF Ziegler’s column, two planar frames, heavy chains with three and many DOF, a continuous model of a simply-supported beam with a cantilever extension and finally a buckled Ziegler’s column. Then, modal asynchronicity is discussed in the non-linear realm: nearly-asynchronous modes in a 3DOF heavy chain and in beams on elastic foundation, one of these with unilateral contact; and exactly-asynchronous modes in a 2DOF heavy chain. To conclude, the technological potentiality of modal asynchronicity for vibration control and energy harvesting is illustrated for a 3DOF heavy chain coupled to a piezoelectric element.

6.1 Introduction Whenever we refer to vibration modes, the first idea that comes to mind is that they are synchronous, which means that all material points undergo motions with the same frequency and phase, so that they all reach their maximal or their minimal displacements simultaneously. Consequently, this vibration mode is a stationary wave. As a rule, modal synchronicity is taken for granted, so much so that when Rosenberg [31] proposed his definition of nonlinear normal modes, he has done it as an

C. E. N. Mazzilli (B) · E. A. R. Ribeiro · B. A. P. Mendes University of São Paulo, São Paulo, Brazil e-mail: [email protected] E. A. R. Ribeiro e-mail: [email protected] B. A. P. Mendes e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. R. Castilho Piqueira et al. (eds.), Lectures on Nonlinear Dynamics, Understanding Complex Systems, https://doi.org/10.1007/978-3-031-45101-0_6

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extension of the classic synchronous linear modes, thus requiring that all material points should still vibrate in unison. When Shaw and Pierre [34] proposed the more embracing invariant-manifold definition of non-linear normal modes, the so-called similar modes to the classic linear synchronous modes were recognized as a particular case, in which the modal invariant manifolds would coincide with the linear system eigenplanes and the non-linearities would be solely embedded in the modal oscillator equation. Shaw and Pierre’s invariant manifold definition helps understanding a possible source of modal asynchronicity, namely phase differences. In fact, by depending on both the generalised field of displacements and velocities, which have an intrinsic phase difference, the ensuing motion might not be synchronous. This is what happens even in the linear analysis if the system has non-proportional damping: it is then seen that the modal motion is a travelling wave [21], depending on the initial conditions. Nevertheless, according to our understanding, it would be desirable that the proposition of an asynchronous-mode definition should be a system property, rather than an initial-condition-dependent event. In other words, modal asynchronicity should be the consequence of frequency differences between the system material point’s oscillations. Different frequency contents should be observable in non-null-measure regions of the system under a generic free-vibration motion, when several modes would take part in the response, provided at least one of them would be asynchronous. Different response spectra in different parts of the system in free vibrations would therefore be indicative of an intrinsic asynchronicity in at least one of its modes. However nice this physical criterion seems to be, it implicitly resorts to Fourier transforms, which are linear operators, and therefore some difficulties are to be expected if we aim at extending the definition of modal asynchronicity to non-linear systems. That is why Lenci and Mazzilli [16, 17] preferred to follow another reasoning, which would not be limited to the linear analysis. In fact, if a vibration mode is localised, that is, if at least one system region is at rest (zero-frequency motion), while other regions are vibrating with a non-null frequency, then there already is a different frequency content in the system response and, hence, modal asynchronicity. In this way, two apparently different phenomena, namely modal asynchronicity and modal localisation, become correlated. In the literature, there is hardly any reference to modal asynchronicity, but several to asynchronicity in general, although with quite different meanings and in quite different situations, mostly in forced dynamics. These are the cases, for instance, of aerodynamic instabilities in blades [36], rotor dynamics and turbomachinery [6, 9, 32, 35, 40, 41]. As for localised modes, although without any reference to its relation with modal asynchronicity, there are quite a few ones published in the last five years of the past century and five first years of the present century [12, 13, 39], also in the broad field of Physics [14]. Surprisingly as it may appear, little was published in the ten following years on modal localisation. Lenci and Mazzilli [16, 17] re-opened the issue, yet establishing the connection between modal asynchronicity and modal localisation. They studied asynchronous modes in the free linear vibrations of the Ziegler’s column, with two, three and four

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DOF, about the trivial and the post-critical equilibrium configuration. Asynchronous modes were also seen to exist in simple portal frames and shear buildings, provided the system mass and stiffness parameters were judiciously chosen. Later studies [19, 20, 23, 30] revealed that asynchronous modes can in fact happen in many other structural systems both discrete and continuous, such as in pre-stressed heavy cables, beams and even in cylindrical shells. Non-linearities can often destroy exact modal asynchronicity, but quasi-asynchronous modes can still appear. There are cases, on the other hand, in which the non-linear system can keep exact modal asynchronicity, as it will be shown in Sect. 6.5 of this Chapter. Although a rare and a non-robust finding, from the structural stability standpoint, modal asynchronicity should be expected, in particular, in the so-called periodic structures [5, 10, 27]. The practical importance and potential technological application of modal asynchronicity is related to either vibration control, when it is wished that specific parts of a structure would be spared from undesirable motions, or to more efficient energy harvesters and signal-enhanced sensors that would take advantage of the concentration of energy in a smaller part of the system. To start with, let us consider the free linear oscillations of an undamped discrete system, according to the equation Mu¨ + Ku = 0,

.

(6.1)

where .M ∈ R N ×N is the mass matrix, .K ∈ R N ×N is the stiffness matrix and . N is the number of DOF of the system. In Eq. (6.1), both.M and.K are typically symmetric and positive definite, as it happens for conservative systems. Yet, at times, the stiffness matrix can be non-symmetric, for example, in circulatory systems, such as those subject to follower forces caused by internal flow [11, 26], external flow [8] or friction [4]. Here, as in the so-called Beck’s column [15], the relevant applied force is tangent to the beam end. Also, the stiffness matrix can be non-positive definite for systems with static instability caused by either conservative non-gyroscopic forces or centrifugal forces [37], for example. Most of these cases are addressed in the examples that follow. In Eq. (6.1), as usual, over dots mean differentiation with respect to time. Hence, ¨ stand for the displacement and acceleration vectors with respect to the .u and .u equilibrium configuration, and we assume that they are directly related to the physical displacements of the structure. The following Sections will illustrate the calibration of the system parameters for several different structural models, thus allowing for the occurrence of asynchronous modes. Basically, the so-called admissibility conditions will be achieved in the form of an inverse eigenvalue problem, imposing the eigenvector to be localised and determining the system parameters that would make it possible, further respecting the condition that the associated eigenvalue would be positive for the asynchronous mode to be stable. Another reasoning to calibrate the system parameters to obtain asynchronous modes could be thought of. Basically, virtual constraints would be added to the

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system to prevent it from vibrating in certain parts, but the corresponding virtual reactions would be set to zero. This would also lead to the admissibility conditions. This latter procedure is physically appealing, but seems to be more laborious, so that we will restrict ourselves to the former.

6.2 Linear Asynchronous Modes About the Undeformed Equilibrium Configuration 6.2.1 2DOF Ziegler’s Column Under Sub-critical Follower Force The model illustrated in Fig. 6.1 was studied in detail in [16]. It has two massless rigid bars of lengths . L 1 and . L 2 connected by linear rotational springs of stiffness . K 1 and . K 2 , lumped masses . M1 and . M2 and a follower force . P at the free tip. The chosen generalised coordinates are the angles .θ1 and .θ2 . In this case, the mass and stiffness matrices of Eq. (6.1) are given by [ 2 ] L 1 (M1 + M2 ) L 1 L 2 M2 .M = (6.2) L 1 L 2 M2 L 22 M2

Fig. 6.1 2DOF Ziegler’s column

6 Asynchronous Modes of Vibration

159

and [ K=

.

] K 1 + K 2 − P L 1 −K 2 + P L 1 . −K 2 K2

(6.3)

Imposing [ ]T u= 10

(6.4)

.

to be an eigenvector leads to λ=−

.

K2 K1 and P = + K2 L 1 L 2 M2 L1

(

1 1 M1 + + L1 L2 M2 L 2

) ,

(6.5)

where the eigenvalue .λ happens to be negative, indicating that the eigenvector corresponds to an unstable mode. Furthermore, this would not really be an asynchronous mode, since the mass . M2 would still be moving with the same frequency as . M1 (see Fig. 6.2a). Imposing now [ ]T u= 01

(6.6)

.

to be an eigenvector leads to λ=

.

(a)

K2 and P = K 2 L 22 M2

(

1 1 + L1 L2

) .

(6.7)

(b)

Fig. 6.2 (a) .u = [1 0]T unstable eigenvector (not an asynchronous mode), and (b) .u = [0 1]T (a stable asynchronous mode)

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Fig. 6.3 The eigenvalues .λ of .M−1 K

Re(λ)

Im(λ)

λML2 K

PL K

Since the eigenvalue .λ is positive, this is in fact a stable asynchronous mode (Fig. 6.2b). Let us assume that .

M1 = M2 = M,

K 1 = K 2 = K and L 1 = L 2 = L .

(6.8)

Now, Eqs. (6.5) and (6.7) simplify respectively to K K and P = 4 2 L M L

(6.9)

K K and P = 2 . L2 M L

(6.10)

λ=−

.

and λ=

.

Figure 6.3 depicts the eigenvalues .λ = ω 2 of .M−1 K as a function of . P. For . P < 2K /L, they are both positive, so that the two (synchronous) modes are stable; for .2K /L < P < 4K /L, the eigenvalues are complex (with real and imaginary parts), so that flutter happens; for . P > 4K /L, the eigenvalues are both negative, so that divergence takes place. It is worth remarking that a stable asynchronous mode appears only for the bifurcation value of . P = 2K /L.

6.2.2 3DOF Ziegler’s Column Under Sub-critical Follower Force The 3DOF Ziegler’s column of Fig. 6.4, which is an extension of the 2DOF model of Fig. 6.1, was studied in detail in [16]. It is studied to show how the asynchronous behaviour is affected by increasing the system’s dimension. Loads . P2 and . P3 are follower forces, forming angles .θ2 and .θ3 with the vertical direction.

6 Asynchronous Modes of Vibration

161

Fig. 6.4 3DOF Ziegler’s column

Alternatively, our choice of generalised coordinates will now be the physical coordinates .xi (see Fig. 6.4). The linearised equations of motion for free vibrations about the equilibrium configuration .x1 = x2 = x3 = 0 are still given by Eq. (6.1), in which ⎡ ⎤ M1 0 0 .M = ⎣ 0 M2 0 ⎦ (6.11) 0 0 M3 and .K is a .3 × 3 matrix with

K 12

(

) ) ( 1 1 2 K3 1 1 + K 2 + 2 − (P2 + P3 ) + , L1 L2 L1 L2 L2 ( ) ( ) K2 1 1 1 P2 K3 1 P2 + P3 =− + + , = K 21 + − + L2 L2 L1 L2 L2 L2 L3 L2 K3 K 13 = K 31 = , L2 L3 ( ) ( ) 1 K2 1 2 1 1 + K 3 − P3 + K 22 = 2 + , L2 L3 L2 L3 L2 ( ) K3 1 1 P3 P3 =− + , + K 23 = K 32 + L3 L3 L2 L3 L3 K3 K 33 = 2 . L3 . K 11

=

K1 + L 21

(6.12)

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C. E. N. Mazzilli et al.

(a)

(b)

(c)

Fig. 6.5 Three possible asynchronous modes. (a) .u = [1 x 0]T , (b) .u = [1 0 x]T and (c) = [0 1 x]T (all exist, but some may be unstable for the values of the parameters reported in the text)

.u

We remark that the eigenvectors .u = [1 0 0]T , .u = [0 1 0]T and .u = [0 0 1]T exist only for . K 3 = 0, which is not a case of mechanical interest. Also, the asynchronous mode .u = [1 x 0]T , where .x is a free parameter (see Fig. 6.5a), has no interest, since the associated eigenvalue is negative and, therefore, the mode is unstable. Let us investigate the eigenvector (see Fig. 6.5b) [ ]T u= 10x .

.

(6.13)

When . L 1 = L 2 = L 3 = L, . M1 = M2 = M3 = M and . K 1 = K 2 = K 3 = K , the following relationships are found: λ=

.

x +1 λM L 2 = , K x

x 3 + 2x 2 − 4x − 1 P2 L = K 2x(x + 1) 2(x + 2) P3 L = . and p3 = K x +1

p2 =

(6.14)

However, only . p2 and . p3 can be assigned, because .x is a part of the system response. So, eliminating .x between . p2 (x) and . p3 (x) gives .

p2 = −

2 p33 − 17 p32 + 42 p3 − 28 . 2( p3 − 4)( p3 − 2)

(6.15)

Figure 6.6 indicates the loci of points where asynchronous modes occur in the parameter space .( p2 , p3 ). Three branches, two stable and one unstable, can be seen.

6 Asynchronous Modes of Vibration

163

Fig. 6.6 Loci of points p3 ) where the asynchronous mode T .u = [1 0 x] exists

Stable

.( p2 ,

Unstable

p3

10

p2

We note that asynchronous motions exist for every combination of signs of . p2 and p3 , apart from the case . p2 < 0 and . p3 < 0, i.e. when both are tensile forces. This shows the ubiquity of the asynchronous modes, although they are non-robust from the structural stability standpoint, as already pointed out. It is worth noting that for . p3 = 4 and for . p3 = 2, any large enough (positive or negative) value of . p2 gives asynchronous mode. Finally, the conditions of existence for the last non-isolated asynchronous mode (see Fig. 6.5c) .

[ ]T u= 01x

.

(6.16)

are examined. When . L 1 = L 2 = L 3 = L, . M1 = M2 = M3 = M and . K 1 = K 2 = K 3 = K , we obtain: λ=

.

λM L 2 x −2 = , K x

P2 L x 3 − 4x 2 + 4x − 2 =− K x(x − 2) P3 L 2(x 2 − 2x − 1) and p3 = = . K x(x − 2) p2 =

Eliminating .x between . p2 (x) and . p3 (x) now gives / − p33 + 5 p3 − 6 ± p32 − 6 p3 + 8 . p2 = . p3 − 2

(6.17)

(6.18)

This curve is depicted in Fig. 6.7, where again three branches, two stable and one unstable, can be seen. Similar comments of Fig. 6.6 also are applicable to Fig. 6.7. The only difference is that in the range of .2 < p3 < 4 no asynchronous solutions exist. It can be shown that the previous asynchronous modes can coexist in two cases: for . p2 = −1.618 and . p3 = 5.236; and for . p2 = 0.618 and . p3 = 0.764. Yet, in the

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Fig. 6.7 Loci of points p3 ) where the asynchronous mode T .u = [0 1 x] exists

Stable

.( p2 ,

Unstable

p3

10

p2

former case, the eigenvalues associated to the asynchronous modes are negative and, therefore, the modes are unstable. On the other hand, in the latter case all modes are stable, as seen from / 1 K .λ1 = 0.236, ω1 = 0.486 and u1 = [0 1 2.620]T , L M / 1 K .λ2 = 0.618, ω2 = 0.786 and u2 = [1 0 − 2.616]T L M

(6.19)

(6.20)

and

1 .λ3 = 6.854, ω3 = 2.618 L

/

K and u3 = [1 − 1.164 0.568]T . M

(6.21)

They are depicted in Fig. 6.8, in which the first and second modes are recognised to be asynchronous.

6.2.3 One-Storey Portal Frame The 3DOF one-storey planar frame of Fig. 6.9—with two columns, a beam and a linear spring of stiffness .k constraining the beam horizontal displacement .δ(t)—was studied in detail in [17]. The other two DOF are the rotations .ϕ1 (t) and .ϕ2 (t). The system parameters are the masses per unit length .m i , the bending stiffnesses . E Ji and the lengths . L i of each element. The unknowns are ordered in the vector u = [ϕ1 (t) ϕ2 (t) δ(t)]T .

.

(6.22)

6 Asynchronous Modes of Vibration

165

(b)

(a)

(c)

Fig. 6.8 The three modes for . p2 = 0.618 and . p3 = 0.764. Associated circular frequencies seen in Eqs. (6.19) to (6.21) Fig. 6.9 One-storey portal frame

Let us look for the asynchronous modes in which either the left or the right columns are at rest: u = [0 1 0]T or u = [1 0 0]T .

.

(6.23)

To have asynchronous modes such as these, an extra real constraint must be added. Suppose the rotation on the left-hand side of the beam is blocked, replacing the left column by two parallel vertical rods close to each other (“two leaf springs”). Then .ϕ1 (t) = 0 and the resulting 2DOF model is characterized by the following consistent mass and stiffness matrices:

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⎤ m 3 L 33 11m 2 L 22 m 2 L 32 ⎥ ⎢ 105 + 105 210 ⎥ .M = ⎢ ⎦ ⎣ 11m L 2 13m 1 L 1 13m 2 L 2 2 2 + + m3 L 3 210 35 35 ⎡

(6.24)

and ⎡ 4E J

2

⎢ L2 K=⎢ ⎣

.

+

4E J3 L3

6E J2 L 22

⎤ 6E J2 ⎥ L 22 ⎥, ⎦ 12E J1 12E J2 + + k L 31 L 32

(6.25)

the displacement vector being u = [ϕ2 (t) δ(t)]T .

(6.26)

u = [1 0]T

(6.27)

.

For the asynchronous mode .

to exist, it is necessary that

λ=

.

1260 11

(

E J2 m 2 L 42

) and m 3 =

)] ( [ ( 3) 11 E J3 L 42 8 L2 + m2. 3 L 33 3 E J2 L 43

(6.28)

Since the eigenvalue .λ is positive, this is a stable mode. As the left column stays at rest and no side motion takes place, .m 1 and .k play no role in this mode. Hence, even in the absence of the spring, this mode is still possible. For the sake of an example, we consider the following parameters: .

L1 = L2 = L3 = L ,

E J1 = 8E J, E J2 = E J3 = E J, m 1 = m 2 = m and k = 0,

(6.29)

which lead to .m 3 = 19m/3. The resulting eigenvalues and associated eigenvectors (illustrated in Fig. 6.10) become ) ( 1260 E J 2 and u1 = [1 0]T .λ1 = ω1 = (6.30) 11 m L4 and λ =

. 2

ω22

3780 = 257

(

EJ m L4

)

]T [ 3 1 . and u2 = − 4L

(6.31)

6 Asynchronous Modes of Vibration

167

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

0

0.2 0.4 0.6 0.8 1.0

0

0.2 0.4 0.6 0.8 1.0 1.2

(a)

(b)

Fig. 6.10 The two normal modes: (a) .u1 = [1 0]T and (b) .u2 = [−3/4L 1]T

The mode reported in Fig. 6.10a is an asynchronous mode and that of Fig. 6.10b is a standard synchronous mode.

6.2.4 Three-Storey Shear Building The three-storey shear building illustrated in Fig. 6.11 was studied in detail in [17]. Beams 9-13 have lumped masses and infinite flexural stiffness. This is a 3DOF system and the horizontal displacements, chosen as the generalised coordinates, are collected in the vector u = [δ A δ B δC ]T .

(6.32)

.

The mass and the stiffness matrices of this shear building are respectively ⎤ ⎡ MA 0 0 .M = ⎣ 0 M B 0 ⎦ (6.33) 0 0 MC and ⎡ I1 +I2 +I3 ⎢ ⎢ K = 12E p ⎢ ⎣

.

H13

+

I4 (H2 +H3 )3

− I5H+I3 6 2

I4 − (H2 +H 3 3)

+

I5 +I6 H23

− I5H+I3 6 2

I5 +I6 H23

+

I7 +I8 H33

− I7H+I3 8 3

I4 − (H2 +H 3 3)

− I7H+I3 8 3

I4 (H2 +H3 )3

+

I7 +I8 H33

⎤ ⎥ ⎥ ⎥ , (6.34) ⎦

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C. E. N. Mazzilli et al.

Fig. 6.11 Three-storey shear building

where . E p and . Ii stand for the Young modulus and the moment of inertia of the cross section of the columns, respectively. Conveniently, the mass and stiffness matrices can be re-written in the form ⎡ ⎤ 100 .M = m ⎣0 ζ 0 ⎦ (6.35) 00μ and ⎡

⎤ 1 + α + γ −α −γ −α α + β −β ⎦ , .K = k ⎣ −γ −β β + γ

(6.36)

in which the dimensionless parameters .α, .β, .γ, .ζ, .μ can be easily obtained by comparing Eqs. (6.35) and (6.36) with Eqs. (6.33) and (6.34). They represent the stiffness and mass ratios and are all positive. Let us first discuss whether an asynchronous mode such as in Eq. (6.37) can exist: u = [0 x 1]T .

.

(6.37)

6 Asynchronous Modes of Vibration

169

The following admissibility conditions arise: (

k γ λ= .x = − , α m

αβ + αγ + βγ γζ

) and (αμ − γζ)(αβ + αγ + βγ) = 0.

(6.38) Since the eigenvalue .λ must be positive for the mode to be stable, from Eq. (6.38).3 it is required that μ=

.

γζ . α

(6.39)

The reasoning that asynchronous modes can be understood as modes of a modified system with added-on virtual constraints with zero virtual reactions is nicely illustrated here. In fact, if a constraint preventing the displacement .δ A of Fig. 6.11 is added to the system, its reaction would be . R = γkδC + αkδ B = k(γ + αx), for .δC = 1 and .δ B = x. Yet, the imposition of . R = 0, would lead to the same conclusion of Eq. (6.38).1 . Remarkably, the asynchronous mode in Eq. (6.37) does not excite the columns between the ground and the first floor, a welcome “architectural” outcome for buildings on pilotis. Yet, although this asynchronous mode would not be excitable by a resonant imposed foundation motion, simple calculations indicate that its modal participation mass. p would be zero, so that the other modes, when excited by frequencies different from that of the asynchronous mode, could still cause large vibrations of the first storey. Ideally, it would be desirable that localisation would also correspond to a maximised modal participating mass, but this is not possible in the present case, as . p is exactly zero. For the sake of a numerical example, the following parameters are chosen: α = 1, β = 0.5, γ = 1 and ζ = 1,

.

(6.40)

leading to .

x = −1, λ = 2

k and μ = 1. m

(6.41)

The associated eigenvalues, eigenvectors and participating masses are λ = 0.268

. 1

k , u1 = [0.732 1 1]T and p1 = 98.1%, m

λ =2

. 2

and

k , u2 = [0 m

− 1 1]T and p2 = 0%

(6.42)

(6.43)

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C. E. N. Mazzilli et al.

(b)

(a)

Fig. 6.12 The three normal modes: (a).u1 = [0.732 1 1]T , (b).u2 = [0 [−2.732 1 1]T

λ = 3.732

. 3

(c)

− 1 1]T and (c).u3 =

k , u3 = [−2.732 1 1]T and p3 = 1.9%, m

(6.44)

showing that the first (synchronous) mode may still be dominant in the system response of the first floor, although the second mode spares it from vibrating. Figure 6.12 depicts the modes. Other asynchronous modes could also be obtained for other system parameters, imposing that the second or the third floors would be at rest. They are not shown here for brevity, but are discussed in [17].

6.2.5 3DOF Pre-tensioned Heavy Chain The 3DOF model of a pre-stressed heavy chain with transversal-displacement linear springs was initially studied in [19]. Then, the model was improved to also include rotational linear springs [20]. Figure 6.13 depicts this latter system, in which hingedhinged rigid bars of length .l are interconnected and constrained by transversal elastic springs of stiffness .kt and rotational elastic springs .kr that mimic the bars’ flexural stiffness . E I . Lumped masses .m are positioned at the hinges. A tension .T is applied upwards at the top, whereas dead weight forces. P = mg (.g is the gravity acceleration) are applied downwards at the hinges. The matrices of mass and stiffness to consider in Eq. (6.1) are respectively: M = ml2 I,

.

where .I is the .3 × 3 identity matrix, and

(6.45)

6 Asynchronous Modes of Vibration

171

Fig. 6.13 3DOF pre-tensioned heavy chain

⎤ 1 + 5α + 2θ − 5σ −4α − θ + 2σ α 2 1 + 6α + 2θ − 3σ −4α − θ + σ ⎦ , .K = k t l ⎣ −4α − θ + 2σ α −4α − θ + σ 1 + 5α + 2θ − σ ⎡

(6.46)

where α=

.

P T kr and σ = . , θ= 2 kt l kt l kt l

(6.47)

The following types of asynchronous modes can appear: q = [1 0 0]T, qB = [0 1 0]T, qC = [0 0 1]T, qD = [1 x 0]T, qE = [0 1 x]T,

. A

(6.48)

in which .x /= 0. Modes of types A to E are all possible and stable for .α = 0 [20]. Yet, only the last two arise when .α > 0, which is of our particular interest here. For type D modes the natural frequency can be evaluated from the eigenvalue .λ = (mω 2 )/kt such that λ=

.

16α2 + 2θ2 + 5σ 2 + 4α + θ − σ + 12αθ − 23ασ − 7θσ 4α + θ − σ −10α2 − θ2 − 2σ 2 + α − 6αθ + 9ασ + 3θσ , = α

(6.49)

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C. E. N. Mazzilli et al.

while for type E modes we obtain −10α2 − θ2 − 2σ 2 + α − 6αθ + 9ασ + 3θσ α 16α2 + 2θ2 + 2σ 2 + 4α + θ − 2σ + 12αθ − 13ασ − 5θσ . = 4α + θ − 2σ λ=

.

(6.50)

A parametric analysis was performed (Figs. 6.14 to 6.17), for decreasing values of .α. From these figures, it is seen that, as .α tends to zero, a mode of the type D tends to type A and a mode of the type E tends to type C. Curiously, when .α is exactly

4ℓ

4ℓ

3ℓ

3ℓ

2ℓ

2ℓ

ℓ

ℓ

0 -2

-1

0

q

1

2

0 -3 -2

-1

(a)

0

q

1

2

3

(b)

Fig. 6.14 For .α = 0.01, .σ = 0.08: (a) mode D with .θ = 0.13760 and eigenvalue .λ = 0.9234, and (b) mode E with .θ = 0.12449 and eigenvalue .λ = 1.0310

4ℓ

4ℓ

3ℓ

3ℓ

2ℓ

2ℓ

ℓ

ℓ

0 -2

-1

0

q (a)

1

2

0 -2

-1

0

q

1

2

(b)

Fig. 6.15 For .α = 0.001, .σ = 0.02: (a) mode D with .θ = 0.03788 and eigenvalue .λ = 0.9807, and (b) mode E with .θ = 0.01820 and eigenvalue .λ = 1.0215

6 Asynchronous Modes of Vibration

173

4ℓ

4ℓ

3ℓ

3ℓ

2ℓ

2ℓ

ℓ

ℓ

0 -2

-1

0

1

q

0 -2

2

-1

(a)

0

1

q

2

(b)

Fig. 6.16 For .α = 0.0001, .σ = 0.005: (a) mode D with .θ = 0.00979 and eigenvalue .λ = 0.9951, and (b) mode E with .θ = 0.00481 and eigenvalue .λ = 1.0051 4ℓ

4ℓ

4ℓ

3ℓ

3ℓ

3ℓ

2ℓ

2ℓ

2ℓ

ℓ

ℓ

ℓ

0 -2

-1

0

q (a)

1

2

0 -3 -2

-1

0

q (b)

1

2

3

0 -2

4ℓ

4ℓ

4ℓ

3ℓ

3ℓ

3ℓ

2ℓ

2ℓ

2ℓ

ℓ

ℓ

ℓ

0 -2

-1

0

q (d)

1

2

0 -2

-1

0

q (e)

1

2

0 -2

-1

-1

0

1

2

0

1

2

q (c)

q (f)

Fig. 6.17 For.α = 0,.σ = 0.005 and.θ = 0.005: (a) mode D with eigenvalue.λ = 0.9829, (b) mode D with eigenvalue .λ = 0.9971, (c) mode C with eigenvalue .λ = 1.0050; for .α = 0, .σ = 0.005 and .θ = 0.01: (d) mode A with eigenvalue .λ = 0.9950, (e) mode E with eigenvalue .λ = 1.0029, (f) mode E with eigenvalue .λ = 1.0171

zero, a type C mode appears as a companion mode to a type D mode and a type A mode appears as a companion mode to a type E mode. A remarkable feature can also be observed in this system, namely that modal asynchronicity is commonly associated with modal frequency accumulation (sev-

174

C. E. N. Mazzilli et al.

eral natural frequencies close by). Hence, three physical phenomena seem to be closely interrelated, namely: modal asynchronicity, modal localisation and modal accumulation.

6.2.6 NDOF Pre-tensioned Heavy Chain The search of asynchronous modes from an inverse eigenproblem can suitably be carried out for discrete systems with few DOF. In such cases, the quantity of modes to be sought is limited, and one can easily arrive at the admissibility conditions for modal asynchronicity for these systems. However, as one regards systems with more and more DOF, so increase the variety of possible asynchronous modes, and discussing them individually becomes meaningless. Instead, one could depart from the results obtained for the systems with few DOF, which serve as reduced-order models, to anticipate modal asynchronicity in systems with many DOF. In this context, it is now considered a NDOF heavy chain (. N >> 1), which is an extension of the model of Sect. 6.2.5. The logic behind the fine-tuning of this NDOF system is simple: if the 3DOF chain displays asynchronous modes, it means the system parameters are tuned for modal asynchronicity, as far as the reduced-order model is concerned. Regarding that all but the pre-stressing T are distributed properties among the system’s DOF, keeping the finely-tuned distribution ratio for the system with NDOF should approximately lead to asynchronous modes, with shapes and frequencies analogous to those in the 3DOF system. Such a ratio is 3 3 3 (kt )3DOF , (kr )NDOF = (kr )3DOF , (P)NDOF = (P)3DOF , N N N (6.51) for .kt , .kr and . P, and (kt )NDOF =

.

l

. NDOF

=

4 l3DOF N +1

(6.52)

for the individual rods’ length. In order to compare the 3DOF and the 48DOF heavy chains, for example, the following arbitrary parameters are defined: .

T = 19.1595 N, kt,3DOF = 100 N m −1 , kr,3DOF = 0.1 Nm rad−1 , m 3DOF = 1 kg and l3DOF = 0.25 m. (6.53)

Such values correspond to α = 0.016, σ = 0.4 and θ = 0.7664,

.

(6.54)

6 Asynchronous Modes of Vibration

0

175

0

0

q

q (a)

(b)

q (c)

Fig. 6.18 3DOF system’s modes. Frequencies: (a) .ω = 7.82 rad s−1 , (b) .ω = 11.13 rad s−1 and (c) .ω = 15.50 rad s−1

Fig. 6.19 7th mode in the 48DOF system with frequency .ω = 3.74 rad s−1

and they yield the modes of Fig. 6.18, namely a D mode and two nearly E modes. Notice that all these modes are asynchronous for practical purposes (as discussed in the previous subsection, this is expected as.α ≈ 0). The modal analysis of the 48DOF chain was carried out with Ansys® [2], which provided, for instance, the mode of Fig. 6.19. All modes, regardless their frequencies, resemble somehow the shapes of D or E modes. The existence of modes with complex frequencies (first to sixth) indicates that modal asynchronicity is also possibly related to statical instability phenomena. In fact, in many cases admissibility conditions for asynchronous modes in heavy chains not only lead to unstable modes, but also correspond to a buckled configuration. Such an interesting feature is detailed in [28].

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6.2.7 Simply-Supported Beam with a Cantilever Extension Consider the structure illustrated in Fig. 6.20. It is a simply-supported beam of span l with a cantilever extension of length .l2 , with the same uniform distributed mass .ρA and flexural stiffness . E I . Furthermore, at . x = ξl1 it is attached a translational oscillator (TO) of stiffness .k and mass .m. This system’s free dynamics is governed by the PDE . 1

.

E I w '''' + ρAw¨ = 0

(6.55)

along the structure, and the ODE m[w¨ TO + w(ξl ¨ 1 )] + kwTO = 0

.

(6.56)

for the oscillator. In the motion equations above, primes and dots mean differentiation with respect to .x and .t, while .w(x, t) and .wTO (t) are the beam’s and the translational oscillator’s time responses. Notice that the systems discussed in the previous subsections are all discrete. Some of them have distributed properties (e.g.: the planar frames of Sects. 6.2.3 and 6.2.4); but still to characterise their motion it suffices a finite number of generalised coordinates. This is not the case of the present system. Hence, to determine admissibility conditions for modal asynchronicity an inverse eigenproblem is no longer feasible. Instead, we can search for a system’s response .w(x, t) which is non-null within a given finite interval, yet null within another one, and this is achieved by means of finely-tuned source of discontinuities. As discussed in [29], to achieve an exact asynchronous mode in a beam continuous model, sometimes it suffices to fine tune the beam’s parameters; other times, the introduction of additional devices (and their parameters) is required. In the present example, TO plays a fundamental role, as it provides the additional parameters .k and .m, whose calibration allows for exact asynchronous modes. We search for the modal response in the form w(x, t) = φ(x) sin ωt

.

Fig. 6.20 Simply-supported beam with a cantilever extension

(6.57)

6 Asynchronous Modes of Vibration

177

and wTO = φ(ξl1 )

.

mω 2 sin ωt k − mω 2

(6.58)

where.φ(x) is one among the uncountable vibration modes such a system can display, and .ω is the respective natural frequency. If, in addition to the standard boundary and continuity conditions, we impose that at .x = l1 the cross-section rotation is w ' (l1 , t) = 0

.

(6.59)

the vibrations along the cantilever extension will be totally supressed and the modal analysis reduces to the motion of the simply-supported part. It reads φ = C1 cos βx1 + C2 sin βx1 + C3 cosh βx1 + C4 sinh βx1

. 1

(6.60)

for .0 ≤ x1 ≤ ξl1 , and φ = C5 cos βx2 + C6 sin βx2 + C7 cosh βx2 + C8 sinh βx2

. 2

(6.61)

for .0 ≤ x2 ≤ (1 − ξ)l1 , in which β = ω2

.

ρA . EI

(6.62)

In turn, the boundary and continuity conditions to be satisfied are: .φ1 (0) = 0, φ''2 ((1 − ξ)l1 ) = 0,

φ''1 (0) = 0, φ2 ((1 − ξ)l1 ) = 0, φ'2 ((1 − ξ)l1 ) = 0, φ1 (ξl1 ) = φ2 (0), φ'1 (ξl1 ) = φ'2 (0), φ''1 (ξl1 ) = φ''2 (0).

(6.63)

Besides, from the shear force discontinuity caused by the TO it comes the jump condition: φ

. 2

k mω 2 ''' = φ''' 2 − φ1 . k − mω 2 E I

(6.64)

This is therefore a problem of eight unknowns and nine equations. To make it determined, the conditions listed in Eq. (6.63) are chosen and gathered in matrix form: [A]{C} = {0},

.

(6.65)

where .C collects the unknowns .C1 to .C8 and the matrix .A must be singular for nontrivial solutions. From Eq. (6.63) come the asynchronous modes and the respective frequencies of the simply-supported part. In turn, the remaining condition Eq. (6.64)

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Fig. 6.21 Admissibility relation between .m and .k for the 1st asynchronous mode and parameters of Eq. (6.66)

Fig. 6.22 Asynchronous modes for the parameters of Eq. (6.66) Table 6.1 Main properties of 1st to 3rd asynchronous modes -1 .βn .m(kg) .ωana (rad s ) .n

.ωnum (rad

s-1 )

Mode

.1

.0.5236

.66.42

.86.75

.86.74

.3rd

.2

.1.0472

.2.62

.347.02

.346.92

.5th

.3

.1.5708

.0.50

.780.79

.780.55

.8th

gives the admissibility relation between .m and .k, which is depicted in Fig. 6.21 for the first (.n = 1) asynchronous mode (actually it is the third mode, since there are two others with lower natural frequencies) and the following parameters: .

E I = 1.3 × 106 Nm2 , ρA = 13 kgm−1 , l1 = 12 m, l2 = 3 m and ξ = 0.5

(6.66)

Figure 6.22 illustrates the first to the third asynchronous modes obtained for the parameters in Eq. (6.66) and also .k = 3 × 105 Nm−1 . For each of such modes, Table 6.1 gives the value of .β, the oscillator’s finely-tuned mass, the mode number and the natural frequency of the simply-supported part (obtained analytically, and numerically with the aid of ADINA® [1]). For a more detailed discussion on these results, [29] is referred.

6 Asynchronous Modes of Vibration

179

6.3 Linear Asynchronous Modes About the Deformed Equilibrium Configuration 6.3.1 2DOF Ziegler’s Column Under Super-Critical Follower Force Lenci and Mazzilli [16] studied asynchronous modal oscillations of a 2DOF column model about its post-critical (buckled) equilibrium configuration (see Fig. 6.23), with .

M1 = m,

M2 = αm,

K 1 = βk,

K 2 = k and L 1 = L 2 = l.

(6.67)

Notice that the definition of the generalised coordinate .θ2 in Fig. 6.23 (relative rotation between the two rigid bars) is different from that of Fig. 6.1 (total rotation of the end bar). It is assumed that prior to the free oscillations, the system buckles and attains a stable post-critical equilibrium configuration .(θ1 , θ2 ), under the action of conservative loads (force . F applied at the middle hinge and weights .mg and .αmg) and a nonconservative load (follower compressive force . P applied at the free end). Depending on the fine-tuning between . F and . P, the system can freely oscillate asynchronously about the post-critical equilibrium configuration.

Fig. 6.23 Buckled 2DOF Ziegler’s column acted upon by conservative and non-conservative forces

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The post-critical equilibrium configuration is the non-null solution of 1 θ2 + ∆ P sin θ2 = 0, β ) ( θ2 + δαβ sin θ1 + θ2 = 0,

θ + δ(1 + α − ∆ F ) sin θ1 −

. 1

(6.68)

where δ=

.

1 mgl F Pl , ∆F = > 1 + 2α + and ∆ P = . βk mg δ βk

(6.69)

Equation (6.68).2 gives (

θ2 .θ1 = −θ2 − arcsin δαβ

) ,

(6.70)

which relates the two angles in the equilibrium position. The linearised free-vibration matrix equation about the post-critical configuration .(θ1 , θ2 ), in terms of the additional angles .(α1 , α2 ), such that .θ1 = θ1 + α1 and .θ2 = θ2 + α2 , is as indicated by Eq. (6.1), in which [ ] 2 1 + 2α(1 + cos θ2 ) α(1 + cos θ2 ) .M = ml (6.71) α α(1 + cos θ2 ) and [ .K

= βk

) ( 1+δ(1+α−∆ F ) cos θ1 +αδ cos θ1 + θ2 ) ( αδ cos θ1 + θ2

] ( ) αδ cos θ1 + θ2 +∆ P cos θ2 ( ) . 1 β + αδ cos θ1 + θ2

(6.72) The condition for having .u = [1 0]T as an eigenvector is ) ( kβδ cos θ1 + θ2 , ml2 cos θ1 )( ( ) cos θ1 + θ2 1 1 ∆ F = (α + 1) − α+ + . cos θ1 1 + cos θ2 δ cos θ1 λ =

. 1

(6.73)

As the eigenvalue .λ1 is positive, this is a stable mode, although not an asynchronous one, since both masses oscillate in unison. On the other hand, the condition for having .u = [0 1]T as an eigenvector is given by

6 Asynchronous Modes of Vibration

) ( k (αβδ cos θ1 + θ2 + 1), 2 αml ) ( 1 1 ∆P = + + αδ cos θ1 + θ2 . β β cos θ2

181

λ =

. 2

(6.74)

The eigenvalue .λ2 is again positive, which means that this is a stable mode and also asynchronous, since the mass . M1 remains at rest in the equilibrium position .θ1 , whereas the mass . M2 oscillates. The system of Eqs. (6.68), (6.73).2 and (6.74).2 allows for the determination of .θ1 , .θ2 , .∆ F and .∆ P . For the sake of an example, we consider α = 1, β = 1 and δ = 0.5.

.

(6.75)

From Eqs. (6.68), (6.73) and (6.74) we get θ = −0.91431, θ2 = 0.29161, ∆ F = 3.26615, ∆ P = 2.45023, k k 0.20746 and λ2 = 1.40615. (6.76) λ1 = ml2 ml2

. 1

If the physical parameters are m = 5 kg, k = 100 Nm rad−1 and l = 1 m,

.

(6.77)

we obtain ω1 =

.

ω2 =

√ √

λ1 = 2.03694 rad s−1 , synchronous mode

λ2 = 5.30312 rad s−1 . asynchronous mode

(6.78)

Figure 6.24 displays a typical dynamical response, for the initial conditions .α1 = 0.1, .α˙ 1 = 0.0, .α2 = −0.05, .α˙ 2 = 0.0. The response is clearly asynchronous, since the tip mass amplitude spectrum identifies two major peaks (at .ω1 = 2.03694 rad s.−1 and .ω2 = 5.30312 rad s.−1 ), whereas the one at the middle hinge has just one major peak (at .ω1 = 2.03694 rad s.−1 ). This asynchronous response is a linear combination of the two modes of vibration found and illustrates how it is possible to have an overall asynchronous free oscillatory response caused by a single asynchronous mode.

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(a)

(b)

(c)

(d)

Fig. 6.24 Example of asynchronous motion between the masses at mid span and column tip

6.4 Nearly-Asynchronous Non-linear Normal Modes About the Undeformed Equilibrium Configuration 6.4.1 3DOF Pre-tensioned Heavy Chain The same system depicted in Fig. 6.13 is recast here, yet this time considering geometric non-linearities through the kinematic relations between the rotational springs’ angular deformations and the generalised coordinates: ϕi = arcsin (qi − qi−1 ) − arcsin (qi+1 − qi ) ) 1( ∼ (qi − qi−1 )3 − (qi+1 − qi )3 . = 2qi − qi−1 − qi+1 + 6 .

(6.79)

The non-linear equations of motion can be obtained from Lagrange’s equation and are concisely written with index notation up to cubic non-linearities as q '' + βi j q j + γi jmq q j qm qq = 0,

. i

(6.80)

where primes indicate derivates with respect to / τ=

.

kt t, m

(6.81)

6 Asynchronous Modes of Vibration

183

where .β is simply the non-dimensional linear stiffness matrix associated to Eq. (6.46), and γ1111 γ1122 γ1223 γ2111 . γ2122 γ2222 γ2333 γ3122 γ3222

α + θ − 25 σ = 10 3 = 6α + 23 θ − 3σ =α = γ1222 = γ1112 = 4α + θ − 23 σ α − 21 θ + 21 σ = − 13 6 = γ1223 = γ2333

γ1112 γ1123 γ1233 γ2112 γ2123 γ2223 γ3111 γ3123 γ3233

= − 13 α − 23 θ + 3σ γ1113 = 21 α 2 13 = −α γ1222 = − 6 α − 21 θ + σ 1 = −2α γ1333 = 16 α = γ1122 γ2113 = γ1223 = 2α γ2133 = γ1233 3 3 = − 13 α − θ + σ γ 2233 = γ3223 2 2 2 = γ1333 γ3112 = γ1233 = γ1123 γ3133 = γ1113 γ3233 = γ2223 = 6α + 23 θ − 23 σ (6.82)

The non-linear normal modes, following the invariant manifold definition and the procedure proposed by Shaw and Pierre [34], are obtained through the expansions up to cubic non-linearities for each generalised coordinate and velocity in terms of a chosen pair of modal variables .(u, v), and depending on nine coefficients .ain and nine .bin , according to q = X n (u, v) = a1n u + a2n v + a3n u 2 + a4n uv + a5n v 2 a6n u 3 + a7n u 2 v + a8n uv 2 + a9n v 3 ,

. n

qn' = Yn (u, v) = b1n u + b2n v + b3n u 2 + b4n uv + b5n v 2 b6n u 3 + b7n u 2 v + b8n uv 2 + b9n v 3 ,

(6.83)

where .i = 1, . . . , 9, .n = 2, 3 and q = X 1 (u, v) = u and q1' = Y1 (u, v) = v.

. 1

(6.84)

Considering the system parameters to be α = 0.01, σ = 0.10 and θ = 0.038179,

.

(6.85)

then a linear asynchronous mode of type D appears, as seen in [20, 30], with q = x = −0.458 and λ = 0.5705.

. 2

(6.86)

Notice that .q1 = 1 and .q3 = 0 in the linear analysis. When the non-linear modes in Eq. (6.83) are evaluated [30], it is found that q = −0.458272q1 − 0.108607q13 − 0.210165q1 q1' ,

. 2

q3 = −0.002840q13 − 0.007288q1 q1'2 ,

(6.87)

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Fig. 6.25 Non-linear normal mode invariant manifold for .q3

Fig. 6.26 Non-linear normal mode invariant manifold sections for .q3

from which it is seen a small cubic correction to the linear estimates of both.q2 and.q3 . Figures 6.25 and 6.26 depict the invariant manifold of Eq. (6.87).2 , which is tangent to the linear system eigenplanes. The non-linear modes keep a quasi-asynchronous character, provided the modal variables do not take large values. From [30], the non-linear modal oscillator is defined by the following ODE: u '' + ηu + ζu(u ' )2 + ξu 3 = 0,

.

(6.88)

where η = β11 + xβ12 = 0.5705,

.

ζ = β12 a82 + β13 a83 = −0.0257, ξ = β12 a62 + β13 a63 + γ1111 + γ1112 x + γ1122 x 2 + γ1222 x 3 = −0.3173. (6.89) A softening behaviour can be anticipated (see Fig. 6.27), as a result of the particular boundary conditions at the top, namely the imposition of a constant tension, instead of a displacement, which would lead to hardening, as it will be discussed in the next subsection.

6 Asynchronous Modes of Vibration

185

Fig. 6.27 Non-linear mode softening frequency-amplitude curve obtained with numerical integration and with the method of multiple time scales

In Fig. 6.27, the parameter .εa0 is related to the modal variable .u through the following relation: u=

.

√ 1 εa0 ei( ητ +ψ) + O(ε3 ) + c.c., 2

(6.90)

which is obtained from the multiple scales method. Here, .i stands for the imaginary unit and .c.c. means ‘complex conjugate’. The parameter .ε is a scaling factor, and .λ0 = η is associated to the linear natural frequency.

6.4.2 Pre-tensioned Beam on a Winkler Foundation A non-linear continuous model of a pre-tensioned beam resting on a Winkler foundation is now considered, Fig. 6.28b, for which the so-called Bessel-like modes [22] are assumed in part of the beam length (for .0 ≤ x ≤ ξ L), the complementary part supposed to be at rest (for .ξ L ≤ x ≤ L). Of course, rigorously speaking, such an assumption implies a discontinuity in the rotations at .x = ξ L and, therefore, is not exact. Parameters of the continuous system are correlated with those of the corresponding 3DOF model discussed in Sect. 6.2.5 of this Chapter, which is depicted in Fig. 6.28a. The static normal force diagrams are also shown in Fig. 6.28. For a detailed discussion of the model of Fig. 6.28b, reference is made to [23]. The usual Bernoulli-Euler non-linear kinematics for the beam theory and a Hookean material with Young’s modulus . E are assumed to obtain the coupled non-linear equations of motion [22]:

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C. E. N. Mazzilli et al.

(b)

(a)

Fig. 6.28 Pre-tensioned prismatic beam on elastic media subjected to varying normal force: (a) 3DOF model and (b) continuous model

)] 1 '2 ' .ρA u ¨ − EA u + w + p = 0, 2 ) ]' [ ( 1 ρAw¨ + E I w '''' − E A u ' + w '2 w ' + kw = 0. 2 [

(

'

(6.91)

In which .u and .w are the axial and the transversal displacements of a point on the beam axis at coordinate .x; .ρ is the beam material density; . A and . I are respectively the cross-section area and moment of inertia; . p is the uniformly distributed axial load; .k is the Winkler medium stiffness coefficient; over dots mean differentiation with respect to time .t and primes mean differentiation with respect to the axial coordinate .x. The classical approximation of negligible axial inertial forces (.ρAu¨ = 0) is also used, so that the normal forces .T (x, t) are: .

)] [ ( 1 = Tb (t) + px = Tt (t) − pL + px, T (x, t) = E A u ' + w '2 2

(6.92)

where .Tb (t) = T (0, t) and .Tt (t) = T (L , t) are respectively the normal forces at the bottom and top. Integration of Eq. (6.91).1 from .x = 0 to .x = L with .ρAu¨ = 0, and imposition of the boundary condition .u(L , t) = u(L , 0), i.e. constant pre-stressing applied via displacement imposition at time .t = 0, lead to

6 Asynchronous Modes of Vibration

187

pL EA EA u(L , 0) − + px + . T (x, t) = L 2 2L

{

L 0

EA w dx = T (x, 0) + 2L '2

{

L

w '2 dx.

0

(6.93) Finally, from Eqs. (6.91).2 and (6.93), a decoupled equation of motion in the transversal direction is obtained, in which the notation .T (x) = T (x, 0) has been used for short: { E A '' L '2 '''' '' ' w .E I w − T (x)w − pw + kw − w dx + ρAw¨ = 0. (6.94) 2L 0 Single-mode dynamics is next pursued, assuming variable separation in the form: w(x, t) = W (x) sin ωt,

.

(6.95)

where .W (x) is the modal shape and .ω is the modal frequency. Of course, full variable separation cannot be exactly achieved using Eq. (6.95) in the non-linear Eq. (6.94). Yet, an approximate solution is searched making use of a temporal Galerkin projection over a cycle, leading to .

EIW

''''

3E A '' W − T W − pW + kW − 8L ''

'

{

L

W '2 dx − ρAω 2 W = 0.

(6.96)

0

Extending the approach used in [33] to this non-linear problem, the so-called fictitious normal force . N (x) is introduced by means of .

E I W '''' −

3E A '' W 8L

{

L

W '2 dx = −N (x)W '' .

(6.97)

0

To grasp the meaning of this fictitious normal force, it is interesting to evaluate it from its ‘definition’ in Eq. (6.97), when the classical linear modes are used: .

Wn (x) = W0n sin

nπx , L

(6.98)

leading to .

N (x) ∼ = N0n =

( nπ )2 L

E Ieq ,

(6.99)

in which .

E Ieq

/ ) ( I W0n 3 2 and r = . = E I 1 + ηn , ηn = 16 r A

(6.100)

Taking (6.97) into (6.96), an ‘equivalent’ cable equation is obtained, in which the true normal force .T (x) is replaced by .T (x) + N (x):

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.

[T (x) + N (x)] W '' + pW ' + (ρAω 2 − k)W = 0.

(6.101)

Solution of (6.101) can be written in terms of the Bessel functions of first and second kind and zeroth order. Yet, here we shall use a quasi-Bessel solution, which can be obtained from the perturbation procedure explained in [22], according to which the natural frequency and the mode .n ∈ Z+ are given by / ωn =

.

√ )]2 1 [ nπ (√ k + Ttn + Tbn , ρA ρA 2L / Tbn sin (z − z b ) , Wn (x) = 4 Tbn + px √ Tbn + px z(x) = √ √ nπ, Ttn − Tbn √ Tbn z b = z(0) = √ √ nπ. Ttn − Tbn

(6.102)

If it is assumed that a localised mode exists along the bottom part of length .ξ L, where .0 < ξ < 1, the aforementioned quasi-Bessel solution can be adapted in the following way: / ωn =

.

/ Wn (x) =

4

[ √ )]2 1 nπ (√ k + Tξn + Tbn , ρA ρA 2ξ L

Tbn sin (z − z b ) for 0 ≤ x ≤ ξ L , Tbn + px Wn (x) = 0 for ξ L ≤ x ≤ L , √ Tbn + px z(x) = √ √ nπ, Tξn − Tbn √ Tbn z b = z(0) = √ √ nπ, Tξn − Tbn

(6.103)

where.Tξn stands for the sum of.T (ξ L) with. N0n . For the sake of an example, Fig. 6.29 depicts the asynchronous modes (with .n = 2) for the continuous and the 3DOF models, respectively, for the following system parameters: kt = 1.2 × 107 Nm−1 , ρA = 691 kgm−1 , L = 800 m, ξ = 0.75, 7 k = 45000 Nm−2 , l = 200 m, g = 10 ms−2 , (6.104) . kr = 4.8 × 10 Nm, E I = 5.04 × 1010 Nm2 , Tt = 370030 N.

6 Asynchronous Modes of Vibration

189

Fig. 6.29 Asynchronous mode for the continuous and 3DOF models

Imposing the ‘equivalence’ between the 3DOF and the continuous model is by no means trivial. For the uniformly distributed mass and stiffness of the Winkler medium it works all right if they lead to the same resultant values of the 3DOF model. However, the correlation of the beam bending stiffness and the lumped rotational springs’ stiffness is not straightforward. This has been done here by imposing that the natural frequency of the second quasi-Bessel mode of the continuous model of a beam with a span of .0.75L would be approximately the same as that of the 3DOF model (recalling that .0.75L is precisely the part of this model set into vibration in the localised mode). Yet, Fig. 6.29 displays a non-negligible difference in the modal shapes normalised with respect to their maximum amplitude. It is interesting to see the effect that different forms of pre-stressing application have in the post-critical behaviour of the non-linear asynchronous modes. It is recalled that in the 3DOF model it was imposed through a constant axial force in one of the beam ends, whereas in the continuous model, it was imposed through a fixed displacement at the same beam end. These distinctly assumed boundary conditions lead to different behaviour: softening for the former—as discussed in [30] and Sect. 6.4.1–(see Fig. 6.30a), or hardening for the latter—as discussed in [23] (see Fig. 6.30b). Notice that both .εa0 (see Eq. (6.90)) and .η (see Eq. (6.100).1 ) refer to the effect of geometric non-linearity and are related through the following relation: / A . .η = εa0 (6.105) I

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Fig. 6.30 Non-linear frequency-amplitude relation: (a) 3DOF model with force-imposed prestressing at .x = L (.ω0 = 8.06 rad s−1 ), and (b) continuous model with displacement-imposed pre-stressing at .x = L (.ω0 = 8.09 rad s−1 )

A classical modal analysis is carried out for a finite-element model with 97 beam elements and properties consistent with those of the continuous model, using ADINA® [1]. It is remarkable that its first five modes are asynchronous, localisation being more intense in the lower modes, as seen in Fig. 6.31. Moreover, the associated modal frequencies are squeezed in a very small range (from .7.60 to −1 .7.95 .rad s , therefore very close to the second-mode estimations of both the 3DOF and the continuous models). These results suggest that a combination of these five modes of the finite-element model would be able to reproduce the single second mode of the continuous model. As in the 3DOF model, modal asynchronicity/localisation appears in a scenario of modal accumulation. Would modal accumulation be a sufficient indicative of modal asynchronicity? It is worth mentioning that modal accumulation has been reported in cylindrical shells [3] and that beams on elastic media, such as those addressed in this Subsection, are lower hierarchy models for these structures, as far as static axisymmetric behaviour is concerned, resorting to a classical analogy [38].

6.4.3 Beam on a Winkler Foundation with Unilateral Contact The continuous model of a beam resting on a Winkler medium with unilateral contact (see Fig. 6.32) was addressed in a congress paper [18] and the main results are summarised here. This problem bears remarkable similarities with that of the vibrations of a steel catenary riser (SCR) resting partially on the seabed. Even if the linearised equations of motion (6.106) for the seabed-detached and the Winkler-supported parts of the beam are used, the moving boundary condition associated with the so-called touchdown point (TDP) pushes us towards the non-linear analysis.

6 Asynchronous Modes of Vibration

(a)

191

(b)

(d)

(c)

(e)

Fig. 6.31 Nearly-asynchronous modes obtained with ADINA® : (a) mode 1 (.ω = 7.60 rad s−1 ), (b) mode 2 (.ω = 7.75 rad s−1 ), (c) mode 3 (.ω = 7.83 rad s−1 ), (d) mode 4 (.ω = 7.90 rad s−1 ), (e) mode 5 (.ω = 7.95 rad s−1 )

Fig. 6.32 Beam on a Winkler foundation with unilateral contact

E I w '''' + ρw¨ + p = 0 for 0 ≤ x ≤ xc , E I w '''' + μw + ρw¨ + p = 0 for x ≥ xc . .

(6.106)

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C. E. N. Mazzilli et al.

In dimensionless form, Eq. (6.106) can be rewritten as .

1 '''' v + v¨ + 1 = 0 for 0 ≤ y ≤ c(τ ), 4 1 '''' v + v¨ + v + 1 = 0 for y ≥ c(τ ), 4

(6.107)

by virtue of the following variable and parameter definitions: /

μ , EI / μ . τ = βt, and β = ρ

μ . y = αx, c = αxc , v = w, α = p

4

(6.108)

It is possible to convert this problem into one with fixed boundaries through a convenient variable transformation [7] .

z=

y − 1, c(τ )

(6.109)

so that u(z, τ ) = v(y, τ ).

(6.110)

.

Nevertheless, the piecewise linear equations (6.106) become the highly non-linear equation: .

1 '''' ¨ ' + c4 u¨ + c4 + H c4 u = 0, u + (1 + z)c2 c˙2 u '' − 2(1 + z)c3 c˙u˙ ' + (1 + z)(2c2 c˙2 − c3 c)u 4

(6.111)

in which . H (z) is the Heaviside function. The solution of Eq. (6.111) is searched in the form .u(z, τ ) = u(z) + δ(z, τ ), (6.112) in which .δ(z, τ ) stands for the free oscillations about the static solution .u(z). Taking Eq. (6.112) in Eq. (6.111) leads to the perturbation equation: .

1 '''' δ + (1 + z)2 c2 c˙2 u '' + (1 + z)2 c2 c˙2 δ '' − 2(1 + z)c3 c˙δ˙' 4 +(1 + z)(2c2 c˙2 − c3 c) ¨ u ' + (1 + z)(2c2 c˙2 − c3 c)δ ¨ ' +c4 δ¨ + H c4 δ + (c4 − c4 )(1 + H u) = 0, 0

(6.113)

where .c0 , which defines the TDP position at .t = 0, is the solution of .

1 4 2 3 c + c + c02 + c0 − u 0 = 0. 6 0 3 0

(6.114)

6 Asynchronous Modes of Vibration

193

1

1

0,5

0.5

v

u 0

0 -0,5

-0.5

-1

-1 -1

0

1

z

2

0

3

(a)

10

y

20

30

(b)

Fig. 6.33 First non-linear mode for .c0 = 6: (a) in .z-space, and (b) in physical . y-space

1 0 0.5

-0.25

u -0.50

v 0

-0.75

-0.5

-1

-1 -1

0

1

z

(a)

2

3

0

10

y

20

30

(b)

Fig. 6.34 Second non-linear mode for .c0 = 6: (a) in .z-space, and (b) in physical . y-space

The solution of Eq. (6.114) must comply with the boundary conditions δ(−1, τ ) = 0, δ '' (−1, τ ) = 0, δ− (0, τ ) = 0, δ+ (0, τ ) = 0,

.

' ' '' '' ''' ''' δ− (0, τ ) = δ+ (0, τ ), δ− (0, τ ) = δ+ (0, τ ), δ− (0, τ ) = δ+ (0, τ ). (6.115)

The wave .δ(z, τ ) is also imposed not to reflect at .+∞ and to assume finite values. A multiple-scales procedure in time and space was carried out in [18], leading to the non-linear modes of vibration. This is not shown here for brevity, but the resulting first mode is shown in Fig. 6.33a (in .z-space) and Fig. 6.33b (in physical . y-space). Analogously, the second mode is shown in Fig. 6.34. Figures 6.33 and 6.34 both display clearly the nearly localised (therefore asynchronous) character of the modes which are furthermore seen to be non-standing waves.

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6.5 Exactly-Asynchronous Non-linear Normal Modes About the Undeformed Equilibrium Configuration Exactly asynchronous non-linear normal modes about the undeformed equilibrium configuration are illustrated in what follows.

6.5.1 2DOF Pre-tensioned Heavy Chain Figure 6.35 illustrates a 2DOF model that has three massless rigid bars of length .l interconnected by linear rotational springs of stiffness .kr and constrained by linear translational springs .kt ; lumped masses .m and .m(1 + ε), where .0 < ε 0. 1 + λr 2

(8.26)

8 Nonlinear Dynamics of Variable Mass Oscillators

225

The corresponding dimensionless form will be written, μ(ξ) =

.

1 , γ = λl2 > 0, 1 + γξ 2

(8.27)

2γξ ∂μ = −( )2 ∂ξ 1 + γξ 2

(8.28)

such that, μ, (ξ) =

.

and ˙ = dμ = − ( 2γξ ) ξ. ˙ μ(ξ, ˙ ξ) 2 dt 1 + γξ 2

.

(8.29)

Clearly, Eq. (8.24) is a strongly nonlinear system. The following figures show results from simulations in some selected scenarios. We took a hard-spring slightly damped case with dimensionless parameters: .ϑ = 0.5, .κ = 4, .ζ = 0.05, .χ = 0.01, .α = 0.5 and .β = 0.5. ˙ of nine simulation scenarios, considering Figure 8.2 presents phase portraits.(ξ, ξ) that the angular velocity is kept at constant values, so that the system reduces to a single-degree of freedom one. The base case of constant mass, i.e. .γ = 0, is shown in the first column. The stable focus at the origin degenerates to a saddle point as the angular velocity is increased. In fact, in the case of a linear spring, .κ = 0, the bifurcation point, at which the origin would lose stability, is given by the √ critical angular velocity .ω ∗ = Θ˙ ∗ = 1. In the analyzed hard-spring case, .ξ ∗ = ± κ−1 (ω 2 − 1) are the new symmetric foci that emerge for .ω > 1, a classic√result for a homogeneous Duffing oscillator. For the pair .(κ = 4, ω = 2), .ξ ∗ = ± 3/2, as can be perceived in the third line of the first column, in Fig. 8.2. These two foci get close to the origin as the variable mass parameter .γ increases, as can be seen in the second and third columns of the third line of Fig. 8.2. Figure 8.3 illustrates a forced response for the rotating-sliding bid. The system is ˙ Θ, Θ) ˙ = (0.01, 0, 0, 0). A very initially at rest. Initial conditions are chosen as .(ξ, ξ, small initial eccentricity is considered. The applied torque follows a hyperbolic law, .T(t) = 0.005 tanh(0.01t). Notice the interlaced trajectories in the projected phase ˙ and the distinct behaviors of position and angular velocity for the three spaces .(ξ, ξ) values of .γ, as shown in Fig. 8.3. The bid escapes from the origin when .Θ˙ = 1+ , which happens at.t = 200, and tends to reach a steady position after oscillating around the corresponding stable foci. Interestingly, it escapes in the opposite direction for .γ = 100.

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Fig. 8.2 A position dependent variable mass bid sliding on a rotating bar mounted on a ring ˙ for nine scenarios. and fixed by two springs; .μ(ξ) = (1 + γξ 2 )−1 , .γ > 0. Phase portraits .(ξ, ξ) ˙ Angular velocity kept constant at .ω = θ/Ω = 0.5, . 1, . 2. Nondimensional parameters: .ϑ = 0.5, 0 .κ = 4, .ζ = 0.05, .χ = 0.01, .α = 0.5 and .β = 0.5

8.2.2 The Generalized Hamilton’s Principle and the Extended Lagrange’s Equation for a Non-material Volume In 1974, McIver published a seminal article [26] in which the Hamilton’s principle was addressed for systems of changing mass. In fact, McIver’s important contribution is based upon the use of the Reynolds’ transport theorem, ending up with a proper form of Hamilton’s principle in this case. Some years later, in 2002, Irschik and Holl, [18], from momentum considerations and by using the mathematical formalism of fictious particles introduced by Truesdell and Tupin, [45], derived a more general form of the Lagrange’s equation for systems modelled trough the concept of nonmaterial volumes. In essence, the material and the non-material volumes, .V and .Vu ,

8 Nonlinear Dynamics of Variable Mass Oscillators

227

Fig. 8.3 A position dependent variable mass bid sliding on a rotating bar mounted on a ring and fixed ˙ and time histories for by two springs;.μ(ξ) = (1 + γξ 2 )−1 ,.γ > 0. Phase portraits projections.(ξ, ξ) a hyperbolic forced condition .T(t) = 0.005 tanh(0.01t). Three variable mass scenarios, .γ = 0, .20, ˙ Θ, Θ) ˙ = (0.01, 0, 0, 0). Nondimensional parameters:.ϑ = 0.5,.κ = 4, .100. Initial conditions .(ξ, ξ, .ζ = 0.05, .χ = 0.01, .α = 0.5 and .β = 0.5

respectively, coincide at a given instant of time. However, their boundaries, .∂V and ∂Vu , respectively, are animated with distinct velocity fields, as sketched in Fig. 8.4. The non-material volume is filled up with the fictious particles that carry on the same field properties as the material ones. In Isrchik and Holl’s nomenclature, the position of the material particles in an inertial cartesian frame is written .p, whereas .r is reserved for the fictious ones. Let the kinetic energy of the material particles included in .Vu be written: .

{ T =

. u

Vu

1 2 ρv dVu . 2

(8.30)

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Fig. 8.4 Material and non-material volumes

Notice that .Tu = T , since .Vu = V instantaneously. Then, based on momentum considerations and on the mathematical formalism of fictious particles, the most general form of Lagrange’s equation for a non-material volume was obtained by Irschik and Holl, [18]: d ∂Tu ∂T . − u =Qj + dt ∂ q˙ j ∂q j

{

∂ Vu

( ) { 1 2 ∂v ∂u ∂v − · n d∂ Vu − ρv · ρv (v − u) · n d∂ Vu . 2 ∂ q˙ j ∂ q˙ j ∂ q˙ j ∂ Vu

(8.31) In Eq. (8.31).q j , j = 1, ..., n are the generalized coordinates in a finite-dimension configuration space which is, supposedly, able to represent the dynamics of the whole system and. Q j , j = 1, ..., n, are the corresponding generalized applied forces. Notice that the first integral term on the non-material volume boundary in Eq. (8.31) is related to kinetic energy, whereas the second integral term is related to the momentum flux across .∂Vu . Recalling the well-known kinematic identities: .

∂p ∂v = , ∂q j ∂ q˙ j

∂r ∂u = , ∂q j ∂ q˙ j

(8.32)

the Lagrange’s equations of Irshik and Holl may be alternatively written: d ∂Tu ∂T . − u =Qj + dt ∂ q˙ j ∂q j

{

∂ Vu

( ) { 1 2 ∂p ∂r ∂p − · n d∂ Vu − ρv · ρv (v − u) · n d∂ Vu . 2 ∂q j ∂q j ∂q j ∂ Vu

(8.33) On the other hand, McIver’s form of Hamilton’s principle for a non-material volume follows from the usual Reynolds’ transport theorem, [26], and reads, ⎛ ⎞ { {t1 ⎜ ⎟ . (8.34) ⎝δTu + δW − ρv · δp (v − u) · n d∂Vu ⎠ dt = 0, t0

∂Vu

8 Nonlinear Dynamics of Variable Mass Oscillators

where .δW =

∑

229

Q j δq j is the virtual work associated to applied forces. From

j

standard procedures of the variational calculus, see e.g. [13], such that, ) ( ) ( ∂Tu d ∂Tu d ∂Tu δq + − .δTu = , dt ∂ q˙ j j ∂q j dt ∂ q˙ j

(8.35)

recalling the well-known identity, δp =

.

∑ ∂p δq , ∂q j j j

(8.36)

and from the fact that .δq j ≡ 0 at both integration limits in Eq. (8.34), the following form of Lagrange’s equation for non-material volumes was derived by McIver, [26], .

d ∂Tu ∂T − u = Qj − dt ∂ q˙ j ∂q j

{

∂Vu

ρv ·

∂p (v − u) · n d∂Vu . ∂q j

(8.37)

Remarkably, Eq. (8.37) misses the first integral term related to the kinetic energy that appears in Irschik and Holl’s Lagrange’s equation, Eq. (8.31). In other words, McIver’s version of the Hamilton’s principle for non-material volumes is unable to recover the more general form deduced by Irschik and Holl. A generalized form of Hamilton’s principle was then constructed in 2013, by Casetta and Pesce, [4], based upon the same mathematical formalism of fictious particles of Truesdell and Tupin, [45]. This generalized form reads, ⎞ ⎛ { { {t1 1 2 ⎟ ⎜ ρv (δp−δr) · n d∂Vu − ρv · δp (v−u) · n d∂Vu⎠dt= 0. . ⎝δTu +δW + 2 t0

∂Vu

∂Vu

(8.38) Recalling that an identity similar to Eq. (8.36) also holds for the virtual displacements of the fictious particles, .δr, and using the standard procedures of calculus of variations, Irschik and Holl’s general form of the Lagrange’s equation for a nonmaterial volume, given by Eq. (8.33) - or Eq. (8.31) – may be then deduced from the generalized Hamilton’s principle given by Eq. (8.38); see [4]. Notice that the second integral term will not contribute on any subregion .∂Vuk , k = 1, ..., N∂Vu of [ ] 1 2 ρv (δp − δr) · n = 0. Such a condition is satis.∂Vu , wherever the integrand . 2 k ∂V u [ ] fied if . v|∂V k = 0, or/and if . (δp − δr) · n ∂V k = 0. The latter possibility may occur u u if: (i) the virtual displacements of both, the material and the fictious particles, are tangent to .∂Vuk ; (ii) the virtual displacements are both null on .∂Vuk ; or (iii) they are identical to each other. Analogous considerations apply for the second integral term in Eq. (8.38), related to the momentum of the material particles. Similar reasonings

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are valid regarding the integral terms appearing in the two alternative forms of the Lagrange’s equation, Eq. (8.31) and Eq. (8.33). Particularly, the contribution coming from the first integral term in Eq. (8.31) is null wherever the relative velocity of the material particles crossing the non-material volume surface with respect to the velocity of this surface is not dependent on the generalized velocities. In the alternative Eq. (8.33), an equivalent statement holds regarding the dependency on the generalized coordinates. The theoretical discussion made above is not sterile at all, having conceptual and practical consequences. Indeed, in the important class of fluid-structure interaction problems dealing with flexible pipes conveying fluid, this discussion is crucial for a proper derivation of the equations of motion. In this class of problems, a plug flow hypothesis is usually adopted, and the McIver’s form of Hamilton’s principle is applied, as may be seen in the treatise by Paidoussis, [32, 33]. In a recent paper, Kheiri and Paidoussis, [23] showed that the first integral term, related to the kinetic energy of the fluid particles in the Casetta and Pesce’s generalized Hamilton’s principle, [4], given in Eq. (8.38) vanishes at the free mouth of the pipe. In fact, in their formulation the adopted plug flow hypothesis imposes that the intensity of the relative velocity of the fluid with respect to the flexible pipe is constant. In other words, imposes that it does not depend on the time derivative of the generalized coordinates, in that case related to the pipe structure configuration. However, that result is strictly valid if the extensibility of the pipe, and so the corresponding contraction, is disregarded. Otherwise, the relative velocity of the particles would depend on the generalized velocities, by continuity. Those conditions would turn the integral term related to the kinetic energy of the fluid particles a non-null quantity, as can be observed in the Lagrange’s equation in the form given in Eq. (8.31) or (8.33). Moreover, if the velocity of the flow is large, this integral term would be non-negligible, even for very small axial deformations. This sort of very complex problems will not be treated in this chapter and will be left to another text. Instead, a simpler problem will be analyzed: the equation of motion of an oscillating water column inside an open vertical and rigid pipe, piercing the free surface. The Lagrange’s equation for a non-material volume, Equation (8.31)—or (8.33)—will be applied to derive a single degree of freedom model, as an alternative to the use of Eq. (8.2)—or, equivalently, of Eq. (8.4). In this much simpler problem, the integral term dependent on the kinetic energy is shown to be non-null. Example 2—The oscillating water column in an open free surface piercing pipe This is an archetypical problem that arises from many engineering applications. Figure 8.5 shows a sketch of the problem, illustrating a free surface vertical rigid pipe fixed to a pier, as an elementary model to interesting ocean engineering applications, e.g., a moonpool of a floating monocolumn oil production platform, [14]; and a subsystem of a wave energy converter device, [30]. The main objective is to assess the resonance behaviour of the water column inside the pipe when driven by the action of incident waves. Resonance might be of interest, as in the case of the wave energy converter, or not, as in the case of the moonpool, since designed to give special protection to the installation and operation of subsea

8 Nonlinear Dynamics of Variable Mass Oscillators

231

Fig. 8.5 The free-surface piercing open pipe problem

equipment. For, as in [35] we assume a plug flow hypothesis of an incompressible liquid, water, and take the free surface level inside the pipe, .ζ, measured from the initially undisturbed free surface at the outside, as the only generalized coordinate defining the system configuration. The non-material volume that encloses the region of interest is defined by the surface. S : S F ∪ SW ∪ S R . Notice that the free surface,. S F , and the wall surface, . SW , are both material surfaces, while the inlet/outlet section surface, . S R , is a non-material one, across which fluxes of mass, momentum and kinetic energy exist. First, we apply the extended Lagrange’s equations for systems with mass explicitly dependent on position, discussed in Sect. 8.2.1. For, we notice that the liquid mass inside the pipe is a linear function of the generalized coordinate .ζ, m (ζ) = ρA (ζ + H ) ,

.

(8.39)

where .ρ is the water mass density and . A is the hydraulic section area of the pipe. Then, .

∂m = ρA, ∂ζ ˙ m˙ = ρAζ.

(8.40)

The kinetic energy of the system is given by, .

T =

1 ρA (ζ + H ) ζ˙2 , 2

(8.41)

from which, the generalized momentum is derived as, .

∂T ˙ = ρA (ζ + H ) ζ. ∂ ζ˙

Therefore, the left-hand terms in Eq. (8.2) read,

(8.42)

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.

d dt

(

∂T ∂ ζ˙

)

= ρA (ζ + H ) ζ¨ + ρAζ˙2 , −

1 ∂T = − ρAζ˙2 . ∂ζ 2

(8.43)

The Autonomous Oscillating Water Column A discussion must be open herein, on modelling the velocity of the gained/lost mass across the pipe mouth. Indeed, it is expected that the outgoing/ingoing jet decelerate/accelerate quickly within a small distance from . S R . Consider that the external flow produced by the jet may be well represented by a distribution of concentric potential vortex-rings placed over the non-material surface . S R . It is well known that the far field asymptotic behaviour of a vortex ring is that of a dipole, so that the velocity at the central line of the jet decays as .1/r 3 where .r is the distance from . S R , normalized with respect to the radius of the ring; see for instance, [42]. However, we are interested in the near field behaviour of the vertical component of the velocity, i.e., .r |→ 0. It can be shown, see e.g. [36], that the vertical velocity on the axis of a vortex ring behaves like .∼ (1 + r 2 )−1/2 (1 − r 2 (1 + r 2 )−1 ) in the near field. In other words, there is no velocity jump across the mouth (.r |→ 0), as would be expected by simply considering the continuity equation of an incompressible fluid. A simple model for the velocity of the expelled/gained fluid particle might then be stated as ˙ with .k the upward vertical unit vector orienting . Oz. .w = v = ζk, The applied forces are due to the differential pressure across . S R , which, based on the discussion above, would be then composed solely by a hydrostatic term4 . Therefore, from Eq. (8.3), .

Q aζ =

∑

∂ Pi = A ( pex − pin ) k · k = A [(ρg H ) − (ρg (H + ζ))] = −ρAgζ. ∂ζ (8.44)

fi ·

i

Also,

.

Q tζ =

∑

m˙ i wi ·

i

∑ ∂ Pi ∂v ∂v ˙ · k = ρAζ˙2 , (8.45) = m˙ i wi · = mw ˙ · i = m˙ ζk ˙ ˙ ∂ζ ∂ ζ ∂ ζ i

and, finally, .

q

Qζ = −

∑ ( 1 ∂m i

4

2 ∂ζ

) i

(vi )2

=−

1 ∂m ˙2 1 ζ = − ρAζ˙2 . 2 ∂ζ 2

(8.46)

It is worth noting that different considerations were done in [35], leading to a different final form for the equation of motion.

8 Nonlinear Dynamics of Variable Mass Oscillators

233

Collecting terms, from Eqs. (8.43) to (8.46) in Eq. (8.2), the equation of motion reads: 1 1 ρA (ζ + H ) ζ¨ + ρAζ˙2 − ρAζ˙2 = −ρAgζ + ρAζ˙2 − ρAζ˙2 , 2 2

.

(8.47)

which, after all simplifications, reduces to the simple form: ρA (ζ + H ) ζ¨ + ρAgζ = 0.

(8.48)

.

Notice that, interestingly, the terms dependent on.ζ˙2 cancel themselves out exactly. The reader can verify that the alternative and simpler form of the extended Lagrange’s equation, Eq. (8.11), leads to Eq. (8.48), as expected. The same result is obtained from Eq. (8.14). Now we take the generalized Lagrange’s equation written for a non-material volume, as given by Eq. (8.31), applied for this single degree of freedom problem, d ∂Tu ∂Tu = Q aζ + . − dt ∂ ζ˙ ∂ζ

{

∂ Vu

1 2 ρv 2

(

∂v ∂u − ∂ ζ˙ ∂ ζ˙

)

{ · n d∂Vu −

ρv ·

∂ Vu

∂v (v − u) · n d∂Vu . ∂ ζ˙

(8.49) The generalized applied force on the right-hand side (r.h.s.) is the same already deduced and given by Eq. (8.44). The first and second terms appearing on the lefthand side are given by Eq. (8.43). We focus our attention to the second and third terms on the r.h.s., in Eq. (8.49). Recall that .∂Vu = S : S F ∪ SW ∪ S R . Then, since the free surface, . S F , is a material one, .(v − u) SF ≡ 0 and .(p − r) SF ≡ 0, so that both terms are identically null on that surface. On the material surface . SW , .[(v − u) · n] SW ≡0 , and the two terms under analysis are also identically null. On the other hand, recall that, ˙ v = ζk,

.

∂v = k, ∂ ζ˙

(8.50)

n| SR = −k.

(8.51)

and that, on the non-material surface, . S R , .

u| SR = 0,

Therefore, the integral terms on . S R reduce to: .

1 2

(

{ ρv2 SR

∂v ∂u − ∂ ζ˙ ∂ ζ˙

{ −

ρv · SR

)

1 · n dS R = − ρAζ˙2 , 2

∂v (v − u) · ndS R = ρAζ˙2 . ∂ ζ˙

(8.52)

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Hence, collecting terms and substituting them in Eq. (8.49) it follows: 1 1 ρA (ζ + H ) ζ¨ + ρAζ˙2 − ρAζ˙2 = −ρAgζ − ρAζ˙2 + ρAζ˙2 . 2 2

.

(8.53)

Terms cancel out and the equation of motion reduces to Eq. (8.48) that is hereinafter simplified to: .

(ζ + H ) ζ¨ + gζ = 0.

(8.54)

This simple archetypical problem reassures that McIver’s form of the Hamilton’s principle for non-material volumes is not the most general one and reinforce the importance of the previous conceptual discussion on modelling the dynamics of flexible pipes conveying fluid. Notice, however, that in that case the generalized coordinates describe the configuration of the vibrating pipe, whereas in the oscillating water column problem the generalized coordinate is, itself, the level of the free surface inside the pipe. We proceed with the analysis by taking the undamped and free nonlinear oscillator given by Eq. (8.54) in the following form: ζ¨ + g

.

ζ = 0, ζ > −H. ζ+H

(8.55)

Notice that a singularity would exist at .ζ = −H , i.e., when the meniscus reaches the mouth of the pipe, i.e., the nonmaterial surface. S R , so that the fluid mass inside the pipe becomes null. This is a physical singularity establishing the domain of validity for this simple model. For small oscillations, i.e., .|ζ| min εT ε ,

(8.87)

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where . denotes the mathematical expectation and the difference is given by, ( ) ˙ q) ¨ − Meq q¨ + Ceq q˙ + Keq q . ε = Θ (q, q,

.

(8.88)

Under an ad-hoc assumption of a Gaussian response, the equivalent linear matrices can be obtained systematically as9 , [40], /

\ ∂Θ .Meq = , ∂ q¨ / \ ∂Θ Ceq = , ∂ q˙ / \ ∂Θ Keq = . ∂q

(8.89) (8.90) (8.91)

Note that in the case of asymmetric non-linearities or/and under constant loads, the response can be described in terms of a mean value, .μq , and a random zero-mean q. For such conditions, Eq. (8.85) can be written as: parcel, . ̯ ( ) ( ) ̯ ˙ ̯ M ̯ q¨ + C ̯ q˙ + K μq + ̯ q + Θ μq + ̯ q, ̯ q, q¨ = μQ + Q(t).

.

(8.92)

The mean response can be obtained taking the expectation of Eq. (8.92), ( )> < ˙ ̯ μq = K−1 μQ − Θ μq + ̯ q, ̯ q, q¨

.

(8.93)

and subtracting Eq. (8.93) from Eq. (8.92), an equivalent form for Eq. (8.85) is obtained: ( ) ̯ ˙ ̯ M ̯ q¨ + C ̯ q˙ + K ̯ q + G ̯ q, ̯ q, q¨ = Q(t), ( ) ( ) < ( )> ˙ ̯ ˙ ̯ ˙ ̯ G ̯ q, ̯ q, q¨ = Θ μq + ̯ q, ̯ q, q¨ − Θ μq + ̯ q, ̯ q, q¨ . .

(8.94)

8.3.2 SL Applied to the Water Column Dynamics Forced by Random Free Surface Waves The water column dynamics driven by an incoming monochromatic small amplitude free surface wave is herein taken in dimensional form as, | | | ∂ϕw || 1 . (8.95) . (ζ + H ) ζ¨ + C (ζ + H ) ζ˙ + C V ζ˙ |ζ˙| + gζ = gξ(t) − 2 ∂t |z=−H \ \ / ∂θi ∂Θ = . The following matrix notation is meant: . ∂q ∂q j /

9

8 Nonlinear Dynamics of Variable Mass Oscillators

245

A quadratic damping is added to emulate possible viscous losses at the inlet/outlet section of the pipe, .z = −H . Note that only linear excitation forces are considered in this simpler formulation. Even though there is no constant parcel in the excitation force in Eq. (8.95), the remaining nonlinearities lead to a mean water column displacement, which can be obtained taking the expectation, μζ =

.

1 < ¨> ζζ . g

(8.96)

Subtracting the mean value from Eq. (8.95), the equivalent linear water column dynamics can be written, ( .

| ) ( ) ( ) | H + Heq ζ¨ + H C + Ceq ζ˙ + g + geq ζ = gξ(t) − ϕwt |

z=−H

,

(8.97)

where /

\ ∂ ( ¨) (8.98) ζ ζ = μζ , ∂ ζ¨ \ \ / / ( ) 1 C V 8 1/2 C ∂ ( ˙) 1 C V ∂ ( ˙ || ˙||) C σζ˙ , (8.99) = ζζ + ζ ζ = μζ + H ∂ ζ˙ 2 H ∂ ζ˙ H 2 H π / \ / \ ∂ ( ¨) ∂ ( ˙) geq = ζζ + C ζ ζ = 0. (8.100) ∂ζ ∂ζ .

Ceq

Heq =

and .σζ˙ denotes the standard deviation of the water column velocity. Note that the quadratic damping term reduced to an equivalent parcel which is linearly proportional to.σζ˙ . Hence, in the frequency domain, the relation between the wave surface elevation and the water column displacement may be written in the linear form, Z(ω) = H(1) (ω)fl(ω),

(8.101)

[ ]−1 H(1) (ω) = ω 2 (H + μζ ) + iω H (C + Ceq ) + g

(8.102)

.

with .

and ( fl(ω) = g 1 + ω

.

Statistical Linearization Procedure

) (h − H )) . cosh (kh)

2 cosh (k

(8.103)

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The determination of the first order response by the statistical linearization method depends on the prior knowledge of the mean water column displacement, which is unknown. Hence, the response can be estimated using an iterative procedure. In this regard, the linear form of the water column dynamics can be used as an initial guess. The next step is to calculate the mean water column displacement and their equivalent coefficients. The values found are compared against the initial guess of each step. The iterative procedure runs until a predetermined criterion is achieved; in this case, by comparing the mean water column displacement and the central moments of displacement and velocity. Examples of iterative procedures can be found in [40]. The pdf of the response offers relevant characteristics of the dynamics under random vibrations and are required to calculate the equivalent linear coefficients. For the statistical linearization case, the response is taken as Gaussian, thanks to the linear relation with the small free surface wave amplitude, supposed Gaussian. Then, ( ) ̯ 1 1 ζ .p( ̯ ζ) = / (8.104) exp − . 2 μ ̯ ζ 2 2πμ ̯ ζ 2 Consider a wave energy spectrum given by. Sξξ (ω). From the standard linear theory the energy spectrum of the water column displacement is then given by, | |2 S (ω) = |H(1) (ω)| Sξξ (ω),

. ζζ

(8.105)

and its second order central moment (the expected value of .μ ̯ ζ 2 ) by, {∞ μ =

Sζζ (ω) dω.

. ̯ 2 ζ

(8.106)

−∞

As any linear process, and explicitly shown in Eq. (8.105), the SL method is only able to approximate the response in the frequency range of the excitation.

8.3.3 Higher Order Procedures: The Statistical Quadratization Actually, although the wave surface elevation is Gaussian, the pdf of the water column displacement will deviate from that. This occurs because, if terms up to second order are consistently retained, force and response cause quadratic transformations on the wave surface Gaussian process. The exact response pdf is unknown. A higher order method, the so-called statistical quadratization procedure, SQ, [12], might be applied. In this method, the equivalent quadratic system is derived from Volterra series expansions retained up to second order terms and the response pdf is approximated

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by using a Gram-Charlier expansion of the Gaussian distribution, [22]. The response pdf can then be approximated in the form, ( ) { } ∂3 1 1 ̯ ζ 1 ̯ / .p(ζ) = 1 − exp − (8.107) μ3 , 6 ̯ ζ ∂ ̯ 2 μ ̯ ζ 2 ζ3 2πμ 2 ̯ ζ

a function of the central moments .μ ̯ ζ 2 and .μ ̯ ζ 3 , which may be shown to be determined by, {∞ μ =

. ̯ 2 ζ

| (1) |2 |H (ω)| Sξξ (ω)dω + 2

−∞

{∞{ −∞

| (2) | |H (ω , ω )|2Sξξ (ω )Sξξ (ω )dω dω , 1 2 1 2 1 2 (8.108)

and {∞{ μ =6

. ̯ 3 ζ

∗

∗

H(1) (ω1 )H(1) (ω2 )H(2) (ω1 , ω2 )Sξξ (ω1 )Sξξ (ω2 )dω1 dω2

−∞

{∞ {{ ∗ +8 H(2)(ω1,ω2)H(2) (ω1,ω−3)H(2) (ω2,ω3)Sξξ(ω1)Sξξ(ω3)Sξξ(ω2)dω1 dω2 dω3 . (8.109) −∞

Finally, the energy spectrum of the water column displacement may be shown to be, | (1) |2 . Sζζ (ω) = |H (ω)| Sξξ (ω) + 2

{∞

| (2) | |H (ω , ω − ω )|2 Sξξ (ω )Sξξ (ω − ω )dω . 1 1 1 1 1

−∞

(8.110) In Eqs. (8.108) to (8.110), .H(2) (ω1 , ω2 ) are quadratic transfer functions (QTF), [12], determined from Fourier transforms of the corresponding time domain quadratic responses, expressed in terms of Volterra series, [1]. Details on their derivations are beyond the scope of this chapter. The reader is directed to [12], for a general analysis and to [43] for a detailed application of the SQ method to the water column oscillator.

8.3.4 Simulations The JONSWAP10 formulation is herein adopted, whose the one-sided spectrum form is given by,

10

Joint North Sea Wave Observation Project.

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S (ω) = 320

. ξξ

2 H1/3

T p4

(

ω

−5

) 1950 −4 exp − 4 ω γ A, Tp

(8.111)

with .γ = 3.3, ⎛ ( ) ⎞ ω −1 2 ωp ⎠, A = exp ⎝− √ σ 2

σ = 0.07, if ω < ω p , σ = 0.09, if ω > ω p , 2π ωp = , Tp

(8.112)

where . H1/3 is the significant wave height, .ω p and .T p are the wave peak frequency and period respectively and .γ is the peakedness factor. To assess the reliability of the statistical methods, their results are compared to those directly obtained from a large number of time domain (TD) simulations. The incident wave field is constructed from 100 frequency components for the statistical linearization and time domain models, while 200 frequencies are used for the double-sided spectrum in the SQ formulation. Note that a suitable number of frequency components are necessary to capture the second order effects on the response given by the sum and difference of the frequency components. In this work, the range of frequencies investigated varied from 0 to 2 rad/s. Another important feature is that the spectral response of the system using TD simulations, in particular the second-order effects, are dependent on the random phase used to create the random surface wave elevation. Hence, the TD results are presented using the ensamble average determined from 10 simulations, produced with different sets of the frequency components random phases. The spectral response of the water column displacement using nonlinear TD simulations was calculated using the pwelch function in MATLABTM . Illustrating the results of those techniques, a free surface piercing pipe with a draught of 15 m is considered, acted by waves with significant height of 4.5 m and a spectrum peak period of 16 s. The depth at the location is hypotetically taken as 320 m. Figure 8.10 shows the energy spectrum. The linear viscous damping coefficient is chosen .C = 0.02s−1 , and the quadratic one as .C V = 0.5. The undamped natural √ frequency of the water column calculated from the linearized system is .Ω0 = g/H ∼ = 0.81rad/s, with corresponding natural period .T0 = 2π/Ω0 = 7.8s. Figure 8.11 shows the water column elevation energy spectrum, determined from time domain simulations compared to those determined by using the SL or the SQ schemes. Note that the SL is able to recover quite precisely the response spectrum at large energy densities, within the excitation range, as meant to be. On the other hand the SQ method is able to recover the TD response spectrum in almost the whole

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Fig. 8.10 The one-sided JONSWAP wave energy spectrum used in the simulations; . H1/3 = 4.5m; . T p = 16s

Fig. 8.11 The water column elevation energy spectrum, determined from time domain simulations (TD) and with the statistical linearization (SL) and quadratization (SQ) schemes; . H1/3 = 4.5m; . T p = 16s

analyzed frequency range, even capturing details due to the second order effects. The first peak frequency in the response spectrum coincides with the wave energy’s one, .ω p ∼ = 0.39rad/s. The second peak appears as consequence of a 1:1 resonance, at a region close to the natural frequency of the linearized system, .Ω0 ∼ = 0.81rad/s. The third and smaller peak at the right denotes a 2:1 resonance, caused by quadratic terms, and related to frequency-sums combinations. The energy at small frequencies are related to slow displacements, resulting from slow quadratic demodulations, coming from frequency-differences combinations. Likewise, Fig. 8.12 shows the probability density function determined from time domain simulations compared to the SL and SQ assessments. Both recover the TD pdf around the mean value. Nevertheless, the SL method loses accuracy at larger water column elevations.

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Fig. 8.12 The probability density function of the water column elevation, determined from time domain simulations (TD) and with the statistical linearization (SL) and quadratization (SQ) schemes; . H1/3 = 4.5m; . T p = 16s

8.4 Conclusion This chapter addressed a somewhat specific topic in nonlinear dynamics. A class of variable mass oscillators was treated formally, by revisiting and discussing the general extended Lagrange’s equations derived for mechanical systems with mass depending on time and position. Newer and simpler forms of such equations were derived and interpreted. An archetypical problem of a variable mass bid sliding on a rotating bar was studied as an example. Then the general Lagrange’s equations for non-material volumes were readdressed. Both formalisms were applied to the classic water column dynamics problem and a single degree of freedom model was analysed, in homogeneous and forced forms. A discussion on modelling the forcing function due to incoming small amplitude free surface waves was given. Finally, by using statistical methods the response of the water column to random free surface waves was assessed and exemplified. Acknowledgements The first author acknowledges the support of the São Paulo State Research Foundation (FAPESP), under the project 2018/16829-8 and of the Brazilian Scientific Research Council (CNPq), research grants nr. 308230/2018-3 and 307995/2022-4.

References 1. Bedrosian, E., Rice, S.: The output properties of volterra systems (nonlinear systems with memory) driven by harmonic and gaussian inputs. Proceedings of the IEEE 59(12), 1688– 1707 (1971). https://doi.org/10.1109/proc.1971.8525 2. Casetta, L.: The inverse problem of Lagrangian mechanics for a non-material volume. Acta Mechanica 226(1), 1–15 (2014). https://doi.org/10.1007/s00707-014-1156-7 3. Casetta, L., Irschik, H., Pesce, C.P.: A generalization of Noether’s theorem for a non-material volume. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik 96(6), 696–706 (2015). https://doi.org/10.1002/zamm.201400196

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Chapter 9

Generalized Krylov-Bogoliubov Method for Solving Strong Nonlinear Vibration Livija Cveticanin

Abstract In this paper an analytic procedure for solving strong nonlinear differential equation of the vibration is developed. The Krylov–Bogoliubov method, introduced for weak nonlinear equations, is generalized for the strong nonlinear ones with power order nonlinearity. Based on the exact solution of the truly nonlinear differential equation (without linear term), given in the form of the Ateb function, the approximate solution of the perturbed version of vibration equation is obtained. In addition, the method is applied for strong nonlinear oscillator with time variable parameter. As the special case, motion of the strong nonlinear oscillator with slow variable mass is investigated. In the paper the steady state motion of the forced strong nonlinear one-degree-of-freedom oscillator is also analysed. Using the Melnikov criteria the condition for existence of deterministic chaos in excited strong nonlinear oscillator with linear damping is determined. For suppressing of the chaos the delayed feedback control, i.e., the ‘Pyragas method’ based on the idea of the stabilization of unstable periodic orbits embedded within a strange attractor is suggested.

9.1 Introduction In general, the oscillator is a system whose motion is repeated periodically with period .T [20]. Such systems and oscillatory motion are evident in nature but also in significant number of artificial and engineering constructions. Usually, the oscillator is modeled as a mass-spring system where the elastic property of the spring is assumed to be linear or nonlinear. The linear oscillator is described with linear differential equation for which the well-known exact analytic solution exist. However, if the spring is with nonlinear property the mathematical model is a nonlinear differential equation. Unfortunately, to obtain the closed form solution for all of the nonlinear differential equations is not possible. Because of that approximate solving procedures for solving nonlinear differential equations are developed [6]. L. Cveticanin (B) Obuda University, 1034 Budapest, Becsi ut 96/B, Hungary e-mail: [email protected]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. R. Castilho Piqueira et al. (eds.), Lectures on Nonlinear Dynamics, Understanding Complex Systems, https://doi.org/10.1007/978-3-031-45101-0_9

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The nonlinear oscillators are usually divided into two groups: oscillators with weak nonlinearity and oscillators with strong nonlinearity. Motion of the oscillator with weak nonlinearity is described with a linear differential equation perturbed with small nonlinear terms .

x¨ + ω 2 x = εφ(x, x), ˙

(9.1)

where.ε δ cr . .

.

γ δ

( ) Remark: For . γδ cr the distance between the stable and unstable manifolds of the homoclinic point .(0, 0) is zero and the manifolds intersect transversely forming the transverse homoclinic ( ) orbits. The presence of such orbits implies that for certain parameters . γδ > γδ cr the Poincare map of the system has the strange attractor and the countable infinity of unstable periodic orbits, an uncountable set of bounded nonperiodic orbits and a dense orbit which are the main characteristics of the chaotic motion.

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9.6.1 Example: Oscillator with Quadratic Nonlinearity For .α = 2 and .εδ = 0.3 the mathematical model of the oscillator is .

x¨ − x + x|x| = −0.3x˙ + 0.220563k cos(t).

(9.119)

Using the Melnikov criteria it is obtained that the . M(t0 ) = 0 for (γ/δ)c = 0.7351.

.

(9.120)

Thus, deterministic chaos may exist if (see [9]) .

γ = 0.7351. δ

(9.121)

For .k = 3, attractor is a complicated set with a fractal structure and represents a strange attractor. In Fig. 9.9a the time history diagram and in b the Poincare mas is plotted. To prove the analytical result the numerical experiment of the system is done. It is obtained that for .α = 2 and .k ∈ [1.3, 1.55] ∧ [2.1, 2.4] ∧ [2.65, 3.4] one of the Lyapunov exponents is positive. For suppressing of the chaos the delayed feedback control, i.e., the ‘Pyragas method’ based on the idea of the stabilization of unstable periodic orbits embedded within a strange attractor is suggested. This is achieved by making a small timedependent perturbation in the form of feedback to an accessible system parameter .

F(t) = K [x(t − τ ) − x(t)],

(9.122)

where . K is a weight of perturbation constant and .τ is the time-delay. The perturbed equation is .

x¨ + δ x˙ − x + x|x|α−1 = γ cos(ωt) + K [x(t − τ ) − x(t)].

Fig. 9.9 a Time history diagram, b Poincare map

(9.123)

9 Generalized Krylov-Bogoliubov Method for Solving Strong …

(a)

279

(b)

Fig. 9.10 a Bifurcation diagram, b Poincare phase plane before (grey line) and after chaos control (black line)

For the suggested example (9.119) the perturbed equation is .x ¨

+ 0.3x˙ − x + x|x|α−1 = 0.220563 · 3 · cos(t) + K [x(t − π) − x(t)],

(9.124)

K = 0.5, t = π.

(9.125)

where .

In Fig. 9.10a the bifurcation diagram and in b the Poincare phase plane before and after chaos control are plotted. It is obvious that for (9.125) the chaos is eliminated.

9.7 Concluding Remarks In this paper the analytic procedure for solving strong nonlinear vibration of the onedegree-of-freedom oscillator is introduced. The method is applied for truly nonlinear equation (without linear term), but also for strong nonlinear equation with additional weak nonlinear terms. As an example, the Van der Pol oscillator is considered. The method is adopted for solving strong nonlinear differential equation with slow time variable parameters. The oscillator with slow mass variation is investigated. The approximate solution for the forced vibration of the nonlinear oscillator is also computed. Using the Melnikov theorem the condition for chaotic motion is obtained. Even the parameter for chaos suppression is determined. The final remark is that the method suggested in the paper uses the Ateb (inverse Beta function). In future investigation, approximate solution based on the generalized trigonometric function has to be done.

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References 1. N.N. Bogoliubov and J.A. Mitropolski. Asimproticheskie methodi v teorii nelinejnih kole-banij. Nauka, Moscow, 1974. 2. W.-H. Chen and R. F. Gibson. Property distribution determination for nonuniform composite beams from vibration response measurements and Galerkin’s method. Journal of Applied Mechanics, 65(1):127–133, mar 1998. 3. L. Cveticanin. Dynamics of machines with variable mass. Gordon and Breach Science Publishers, London, 1998. 4. L. Cveticanin. The approximate solving methods for the cubic duffing equation based on the Jacobi elliptic functions. International Journal of Nonlinear Sciences and Numerical Simulation, 10:1491–1516, 2009. 5. L. Cveticanin. Dynamics of the mass variable body. Chapter 3, in “Dynamics of mechanical systems with variable mass”. Series CISM International Centre for Mechanical Sciences, 2014. 6. L. Cveticanin. Strongly nonlinear oscillators – analytical solutions. Springer, Berlin, 2014. 7. L. Cveticanin. Strong nonlinear oscillators – analytical solutions. Springer, Berlin, 2018. 8. L. Cveticanin, S. Vujkov, and D. Cveticanin. Application of Ateb and generalized trigonometric functions for nonlinear oscillators. Archive of Applied Mechanics, 90(11):2579–2587, 2020. 9. L. Cveticanin and M. Zukovic. Melnikov’s criteria and chaos in systems with fractional order deflection. Journal of Sound and Vibration, 326(3-5):768–779, oct 2009. 10. L. Cveticanin, M. Zukovic, and D. Cveticanin. Exact steady states of periodically forced and essentially nonlinear and damped oscillator. Communications in Nonlinear Science and Numerical Simulation, 78:104895, nov 2019. 11. L. Cveticanin, M. Zukowic, and D. Cveticanin. Steady state vibration of the periodically forced and damped pure nonlinear two-degrees-of-freedom oscillator. Journal of Theoretical and Applied Mechanics, 57(2):445–460, apr 2019. 12. I.S. Gradstein and I.M. Rjizhik. Tablici integralov, summ, rjadov I proizvedenij. Nauka, Moscow, 1971. 13. J. Guckenheimer and P. Holmes. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer, New York, 1983. 14. H. W. Haslach. Post-buckling behavior of columns with non-linear constitutive equations. International Journal of Non-Linear Mechanics, 20(1):53–67, jan 1985. 15. H. W. Haslach. Influence of adsorbed moisture on the elastic post-buckling behavior of columns made of non-linear hydrophilic polymers. International Journal of Non-Linear Mechanics, 27(4):527–546, jul 1992. 16. I. Kovacic, L. Cveticanin, M. Zlokolica, and Z. Rakaric. Jacobi elliptic functions: A review of nonlinear oscillatory application problems. Journal of Sound and Vibration, 380:1–36, 2016. 17. G. Lewis and F. Monasa. Large deflections of cantilever beams of non-linear materials of the ludwick type subjected to an end moment. International Journal of Non-Linear Mechanics, 17(1):1–6, jan 1982. 18. V.K. Melnikov. On the stability of the center for time periodic perturbations. Transactions of the Moscow Mathematical Society, 12:1–57, 1963. 19. R.E. Mickens. Truly nonlinear oscillations. World Scientific, 2010. 20. A.H. Nayfeh and D.T. Mook. Nonlinear oscillations. Wiley, New York, 1979. 21. G. Parthap and T.K. Varadan. The inelastic large deformation of beams. Journal of Applied Mechanics, 45:689–690, 1976. 22. W.N. Patten, S. Sha, and C. Mo. A vibrational model of open celled polyurethane foam automative seat cushion. Journal of Sound and Vibration, 217(1):145–161, oct 1998. 23. R.M. Rosenberg. The Ateb(h) functions and their properties. Quaterly of Applied Mathematics, 21:37–47, 1963. 24. D. Russel and T. Rossing. Testing the nonlinearity of piano hammers using residual shock spectra. Acoustica-Acta Acustica, 84:967–975, 1998.

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Chapter 10

Chaos Theory Marcelo A. Savi

Abstract This chapter presents an overview of chaos theory. It starts from the background of dynamical systems, presenting the mathematical representation and the concept of stability. Afterward, chaotic dynamics is explored from the horseshoe transformation, establishing that it is a consequence of the contraction-expansionfold process. The main aspects of chaotic behavior are then discussed defining chaotic and fractal attractors. Routes to chaos are investigated showing some definitions of bifurcation, treating local and global bifurcations. Lyapunov exponents are defined in order to present a diagnostic tool for chaos.

10.1 Introduction Nonlinearities are the essential characteristics responsible for a great variety of possibilities of natural systems. Rhythms are dynamical manifestations of the natural system behavior with an intrinsic richness expressed by regular and irregular responses over time and space. These ideas motivate dynamical investigations in different areas of human knowledge, varying from mechanics to biology. The scientific revolution is symbolic represented by Galileo Galilei (1564–1642) who introduced the idea of the experimental verification as the main source of the truth and the mathematics as the alphabet of the universe. Isaac Newton (1643– 1727) consolidated these revolutionary ideas establishing the laws of motion and the mathematical description of the phenomena from governing equations. During this scientific revolution, nonlinear systems were usually avoided creating a linear paradigm that limited the human comprehension about natural processes. One of these paradigms is the strict determinism, clearly illustrated by the PierreSimon Laplace (1749–1827) thinking: “If we conceive of an intelligence which at a given instant comprehends of all the relations of the entities of this universe, it could

M. A. Savi (B) Center for Nonlinear Mechanics—Department of Mechanical Engineering—COPPE—Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. R. Castilho Piqueira et al. (eds.), Lectures on Nonlinear Dynamics, Understanding Complex Systems, https://doi.org/10.1007/978-3-031-45101-0_10

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state the respect positions, motions, and general effects of all these entities at any time in the past or future”. Philosofically, the strict determinism questioned the free will, establishing that nothing can be changed after the definition of the “relations of the entities of the universe” or, in other words, the governing equations. The strict determinism and the linear paradigm started to be broken only in the end of the XIX century. Motivated by the stability analysis of the universe, Jules Henri Poincaré (1854–1912) studied the dynamical response of the three-body problem. Poincaré presented a counterpoint for the strict determinism of Laplace: “Even if the case that the natural laws had no longer secret for us, ... it may happen that small differences in initial conditions produce very great ones in the final phenomena”. This is the essential characteristic of nonlinearity that means that small causes may generate great effects. Chaos is a possible kind of response of a nonlinear system being characterized by sensitive dependence on initial conditions. Hence, a nolinear system has a rich dynamic including regular and irregular behaviors. Although Poincaré has an absolutely clear vision with respect to chaos (as it is understood nowadays), only in 1963, when Edward Lorenz (1917–2008) developed meteorology studies, this idea came back to the scientific scenario [11]. The Lorenz’s analysis associated the sensitive dependence of initial conditions on the idea of the butterfly effect, which means that if a butterfly flaps its wings in China it may cause a hurricane in Brazil. Afterward, nonlinear dynamics and chaos concepts started to be incorporated for the proper investigation of several subjects, being applied in all areas of human knowledge passing through engineering, mechanics, chemistry, biology, economy, psychology, among others. This new approach represented the dynamical freedom, bringing for the analysis ideas hidden by the linear thinking ([4, 10, 13, 18, 22]). This chapter presents a general overview of the chaos theory, providing a formal comprehension of the chaotic dynamics. Initially, dynamical systems background is provided, presenting their mathematical representantion and the concept of stability. The formal definition of chaos is treated in the sequence using the idea of the horseshoe transformation. Chaotic and fractal attractors are discussed. Lyapunov exponents are then presented as a diagnostic tool of chaos.

10.2 Dynamical Systems: Background A dynamical system is a frame-by-frame description of reality being represented by a transformation. f imposed to state variables.x—employed to describe a phenomenon, defining a vector field. Its mathematical description is a set of differential equations as follows: .

x˙ = f (x), x ∈ Rn .

(10.1)

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Fig. 10.1 Dynamical system = f (x), .x ∈ Rn represented by a frame-by-frame description showing how an object evolve to a new one

. x˙

This system is called autonomous since it does not have an explicit dependence on time. On the other hand, it is possible to consider a non-autonomous system that has an explicit time dependence as follows: .

x˙ = f (x, t), x ∈ Rn .

(10.2)

A non-autonomous system can be represented by an autonomous system by increasing the system dimension. Dynamical system analysis can be performed by a geometrical perspective, usually called topology, studying continuous transformations. In this regard, consider that the space of dependent variables, .x, called state space or phase space, has different topologies, characterizing several dynamical aspects. An interesting approach to understand the system dynamics is to monitor an object in phase space, observing how it evolves under transformations imposed by . f . This object can be built by considering different initial conditions and, for each one of these conditions, there is an orbit or trajectory, evolving from one frame to the subsequent ones. These trajectories are the solution of the differential equations and, based on this, a new object is formed in a subsequent specific frame. Figure 10.1 presents the frame-by-frame description showing an object on one frame evolving for a new object in the new frame. Since nonlinear systems usually do not have analytical solution, they need to be treated with proper tools. Numerical procedures are usually employed for this aim. The fourth-order Runge-Kutta method is an example of a numerical procedure to obtain the desired solution ([19]). Perturbation techniques are also an interesting alternative to solve nonlinear systems ([14]). Special procedures should also be employed to treat unstable solutions since the so-called brute force integration is capable to capture just the stable solutions.

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(b)

(a)

Fig. 10.2 Stability concept of Lyapunov. a Stable; b asymptotically stable

10.2.1 Stability Stability is an essential issue of nonlinear dynamical systems being associated with characteristics of a solution subjected to perturbations. If a perturbation does not affect a system response in a significant way, the system is stable. Otherwise, the system is unstable. Alexander Lyapunov (1857–1918) developed a stability theory for dynamical systems establishing a relationship between a specific orbit or solution and its perturbation, represented by a nearby orbit associated with different initial conditions in the neighborhood of the original one. The stablity concept of Lyapunov defines a stable system in such a way that two nearby orbits remain close to each other with the evolution of time. Figure 10.2 shows the idea of the Lyapunov stability showing that the system is stable if there is a .δ = δ(ε) > 0 in such a way that: If

.

| ( ) ( )| |ψ t − φ t | < δ then |ψ (t) − φ (t)| < ε. 0 0

(10.3)

The system is called asymptotically stable if these orbits tend to converge to each other when time tends to infinity, being defined by .δ > 0 in such a way that (Fig. 10.2b): If

.

| ( ) ( )| |ψ t − φ t | < δ then 0 0

lim |ψ (t) − φ (t)| = 0.

t→∞

(10.4)

10.3 Chaos Nonlinear dynamical systems present a great variety of responses that can be understood as a system freedom, associated with rich behaviors. Chaos is one of these possibilities related to richness and unpredictability. In brief, chaos may be defined as the apparent sthocastic behavior of deterministic systems ([1, 9, 12, 16, 19, 23, 24]).

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Fig. 10.3 Frame-by-frame description of the horseshoe transformation

Since a dynamical system may be understood as a transformation. f that is imposed to state variables .x, it is interesting to imagine a special type of transformation characterized by a sequence of contraction-expansion-fold process that represents an archetypal behavior of the system being called horseshoe transformation. This transmormation was originally proposed by the mathematician Steve Smale, and because of that, it is sometimes called as the Smale horseshoe. In order to understand the horseshoe transformation, consider an object in the phase space—a unitary square—subjected to a contraction-expansion-fold process illustrated in Fig. 10.3 as a frame-by-frame description. As a matter of fact, two different transformations can be imagined: a positive part of transformation, . f , and a negative part, . f −1 , represented in Fig. 10.4. The positive part has the expansion on the vertical direction while the contraction occurs on the horizontal direction. The folding process occurs in the sequence. On the other hand, the negative part is the opposite, presenting the expansion on the horizontal direction and the contraction on the vertical. The folding is similar. The limit as the number of interactions of these transformations tends to infinity, produces a set of vertical lines for the positive part of transformation and a set of horizontal lines for the negative part. It is noticeable that these lines can be identified by a sequence of 0’s and 1’s. An invariant set of points can be built by the intersection of positive and negative parts of the horseshoe transformations. Since a generic point of this invariant set is an intersection of vertical and horizontal lines, it can also be identified by a sequence of 0’s and 1’s. Because of that, it is possible to build a structure that represents orbits of dynamical systems from these binary sequences. This approach is called symbolic dynamics and, since it is based on sequences of integer numbers, it is not associated with floating point errors, being useful in several situations. Symbolic space has a

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Fig. 10.4 Horseshoe transformation associated with contraction-expansion-fold process Fig. 10.5 Two points on the invariant set associated with the horseshoe transformation

topological equivalence with the real one, and therefore, it is possible to analyze the system dynamics from the symbolic space. In order to establish an interpretation of the horseshoe transformation, it should be pointed out that each one of the effects of the contraction-expansion-fold process represents three distinct directions respectively associated with stable, unstable and neutral chararacteristics. In this regard, there is an unstable direction that promotes a divergence between two close points. This unstable direction characterizes the sensitive dependence on initial conditions. On this basis, consider two generic points on the symbolic space, . p1 and . p2 (Fig. 10.5) that belong to a small neighborhood, .ε. These points are associated with counterparts on the real space. Due to the characteristics of the horseshoe transformation, it does not matter how close these points are, or in other words, how small is the neighborhood, there is a finite number of iterations that makes these points to be separated by a finite distance. This means that two close points on phase space have sensitive dependence on initial conditions. The existence of the horseshoe transformation is the fundamental dynamical characteristic of chaos. Therefore, chaos needs to be associated with nonlinear systems with, at least, three dimensions, related to distinct directions: contraction, expansion

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Fig. 10.6 Schematic representation of the sensitive dependence on initial conditions associated with horseshoe transformation

and a neutral direction, where fold occurs ([6, 19, 25]). Due to the unstable direction, associated with expansion process, chaotic behavior has a sensitive dependence on initial conditions, which establishes that small causes are related to great effects, the holistic image of the butterfly effect, Fig. 10.6, presents a schematic picture that shows the connection of the horseshoe transformation and the sensitive dependence on initial conditions. By observing a chaotic behavior in phase space, it is possible to observe open orbits that never closes. Time history of state variables have non-periodic characteristic. Figure 10.7 presents system dynamics of the Lorenz system that is a 3-Dim system that describes the Rayleigh-Benard fluid convection, which contemplates two parallel plates, separated by a fluid, where the upper plate has a lower temperature when compared with the lower plate. Tiny variations on initial conditions causes dramatic changes on system responses, with a clear divergence of nearby orbits.

10.3.1 Chaotic Attractors The horseshoe transformation invariant set has the structure of the Cantor set that is closed, disconnected with an uncountable infinity of points. An essential example of this set is shown in Fig. 10.8 considering a line that is split into three equal parts and the center part is discarded. This process is repeated and, when the process tends to infinity, a disconnected set of points is generated. Since the original line is a typical 1-Dim structure and the set generated by the process has disconnected characteristic ‘-’ that is not a point, it is possible to infer that it has a fractional dimension, between

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Fig. 10.7 Lorenz system showing a typical chaotic orbit represented by the phase space (upper panel), phase subspaces (central panel) and state variable time history (lower panel)

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Fig. 10.8 Cantor set–one dimensional structure built with the rule that considers a line that is split into three equal parts and the center part is discarded Fig. 10.9 Poincaré map

1 and 0. This kind of structure has a fractal characteristic as a reference of the non-integer, fractional nature of its dimension. The visualization of the chaotic behavior is interesting to be performed with the aid of Poincaré map that is a stroboscopic view of the system dynamics. There are several possibilities to build a map but, it reduces the system dimension by a transformation. A secant section in the phase space is a strategy that allows to observe the time instants when orbits transversally cross the section. Another situation is related to a periodic external excitation that defines a frequency of the stroboscopic view. Figure 10.9 presents a geometric view of the idea of the Poincaré map construction. In order to illustrate the physical behavior of the horseshoe transformation, consider an object represented by a circle of initial conditions. After some interactions, it is evaluated by the intersection of the orbits on a Poincaré section. If the system presents a chaotic behavior, it is associated with a horseshoe transformation and therefore presents contraction-expansion-fold process. Figure 10.10 shows the evolution of a circle of initial conditions in different time instants, or different Poincaré sections, considering a chaotic response. Periodic response is presented in Fig. 10.11. Since the system presents a regular, periodic behavior, there is not an expansion process, which means that orbits are convergent.

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Fig. 10.10 Chaotic evolution of a circle of initial conditions

Fig. 10.11 Periodic evolution of a circle of initial conditions

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Fig. 10.12 Chaotic strange attractor–different Poincaré sections

Dissipative dynamical systems are characterized by asymptotic behavior, being associated with attractors. Several types of attractors can be observed in dynamical system. A stable equilibrium point can be understood as a 0-Dim attractor. A limitcycle is another possibility of a 1-Dim attractor. Chaotic behavior is also related to an attractor that represents a preferred region of the phase space where orbits converge. A chaotic attractor represents a collection of points, organized in lamelas, with voids, being associated with a Cantor set produced by the horseshoe transformation. Due to that, the name strange attractor is usually employed where its strangeness is related to a geometrical aspect, essentially fractal, with non-integer dimension. Chaoticity, on the other hand, is a dynamical aspect. Therefore, although not usual, it is possible to have different situations with respect to chaos in dynamical systems: chaotic strange attractor; chaotic non-strange attractor; strange non-chaotic attractor ([5]). A typical chaotic, strange attractor is presented in Fig. 10.12 showing different Poincaré sections. A careful observation of these attractors allows one to see the horseshoe transformation applied to the attractor set of points ([19]). Chaotic behavior has the sensitive dependence on initial conditions as its essential characteristic. But the fundamental structure behind chaos richness is the existence of an infinite number of unstable periodic orbits embedded in the chaotic behavior. This means that the complex chaotic orbit is a consequence of the visit to an infinity

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number of periodic orbits and, since they are unstable, the system visits each one of them, being expelled in the sequence. This characteristic confers flexibility to the chaotic behavior, being one of its essential aspects. The common sense knows this kind of idea when analyses human behavior. Huge changes are easier when your life is a mess, chaotic. A well-established life, with periodic characteristics, is difficult to be changed. This points to the advantages of the use chaos instead of resist to it ([2]).

10.4 Route to Chaos The different responses of a dynamical system are defined by parameters and initial conditions. Each set of parameter produces a specific response, and a proper comprehension of system dynamics includes the form of how system behavior is altered by parameter changes. Poincaré introduced the idea of qualitative changes in solution structure using the term bifurcation. Multistability is a nonlinear characteristic where a specific set of parameters can be associated with more than one stable solution. In these cases, initial conditions define the system behavior. Bifurcation analysis is useful to identify dynamical system qualitative changes, defining the routes to chaos. In general, two types of bifurcation can be imagined: local and global. Local bifurcations are restricted to regions of phase space. On the other hand, global bifurcations are non-local. Local bifurcation analysis is usually developed based on normal forms that represent prototypes of bifurcations ([6, 19, 23, 25]). The local bifurcation analysis can be performed around a bifurcation point, defining possibilities related to dynamical changes. Figure 10.13 shows some classical forms of bifurcations related to the creation and destruction of solutions or equilibrium points. It is noticeable the unstable and stable aspects of each solution, which essentially defines the system response.

Fig. 10.13 Examples of local bifurcations observed in dynamical systems

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Fig. 10.14 Global bifurcation represented by two different set of parameters

Global bifurcations are related to qualitative changes in global system aspects and cannot be observed from local analysis. In essence, a parameter change can cause a global change in the orbit structure. This type of bifurcation can explain the birth of chaos due to some orbit collision, for instance. Horseshoe transformation starts due to the transversal touch between unstable and stable manifolds. There is a specific set of parameters that causes this situation, which is responsible for the chaos birth. Figure 10.14 shows unstable and stable manifolds for two different set of parameters. Note that the parameter variation promotes a touch of these manifolds, causing the birth of horseshoe transformation and, consequently, the chaos. For more details, see [6, 7, 19, 23, 25]. Bifurcation diagrams constitute an important tool to identify the influence of parameter changes in system response. It represents a stroboscopically sample of a system variable under the slow quasi-static change of a system parameter. This can be built by a brute force method, just simulating solutions for different parameters. It is important to discard transient response and to define the initial conditions for each parameter. In general, there are two possibilities for the initial conditions: reset for each parameter; use the last simulation as initial conditions. A typical bifurcation diagram is presented in Fig. 10.15 ([19]). Unstable solutions are not captured by brute force integrations, needing proper algorithms for that ([1, 17]).

10.5 Lyapunov Exponents Chaotic behavior needs to be properly identified and diagnostic tools are essential for this aim. Attractor dimension and Lyapunov exponents are dynamical system invariants usually employed to identify chaos. Lyapunov exponents evaluate the sensitive dependence on initial conditions estimating the local divergence of nearby orbits. These exponents have been used as the most useful diagnostic tool for chaotic system analysis and can also be used for the calculation of other invariant quantities as the attractor dimension.

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Fig. 10.15 Bifurcation diagram

In order to understand the idea related to the determination of Lyapunov exponents consider a . D-sphere of states that is transformed by the system dynamics in a . D-ellipsoid. Lyapunov exponents are related to the contraction-expansion nature of different directions in phase space. The evaluation of the divergence of two nearby orbits is done considering the relation between the initial . D-sphere and the . Dellipsoid (Fig. 10.16). It should be highlighted that, as a matter of fact, there is a spectrum of Lyapunov exponents representing all system dimensions. Based on that, these exponents are expressed by: d(t) = d0 bλt ,

.

(10.5)

where .d is the length, .b is a reference basis, and .λ is the Lyapunov exponent. Hence, there is a Lyapunov spectrum given by: ( ) d(t) 1 . logb (10.6) .λ = t d0 The signs of Lyapunov exponents provide a qualitative picture of the system dynamics. The existence of positive Lyapunov exponents defines directions of local instabilities and any system containing at least one positive exponent presents chaotic behavior. A response with more than one positive exponent is called hyperchaos ([3, 20]). Negative or null Lyapunov exponent are associated with trajectories that do not diverge. In addition to the signs of the Lyapunov exponents, their values also bring important information related to system dynamics. Since the exponents evaluate the

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Fig. 10.16 Lyapunov exponents

average divergence of nearby orbits, dissipative systems have a negative sum of the whole Lyapunov spectrum. The determination of Lyapunov exponents of dynamical system with an explicit mathematical model, which can be linearized, is well established from the algorithm proposed by [26]. On the other hand, the determination of these exponents from time series is quite more complex. In essence, there are two different classes of algorithms: trajectories, real space or direct method; and perturbation, tangent space or Jacobian matrix method ([3, 8, 19, 26]). Since chaotic situations are related to local exponential divergence of nearby orbits, it is necessary proper algorithms in order to evaluate Lyapunov exponents ([17, 26]). These algorithms evaluate the average of this divergence considered in different points of the trajectory. Hence, when the distance .d(t) becomes large, it is defined a new .d0 (t) in order to evaluate the divergence, as follows (Fig. 10.17): λ=

.

) ( n 1 ∑ d(tk ) logb . tn − t0 k=1 d0 (tk−1 )

(10.7)

The new value for the perturbed orbit is defined from de Gram-Schmidt normalization. Besides that, the perturbed system can be monitored by different ways ([15]): linearized system, evaluated from the the Jacobian matrix of the system ([26]); cloned dynamics, evaluated by a clone of the the governing equations ([21]). Lyapunov exponents can be employed to calculate other system invariants as attractor dimension. The Kaplan-Yorke conjecture establishes a way to calculate attractor dimension from the spectrum of Lyapunov exponents ([19]): ∑j .

D= j+

i=1 λi , |λ j+1 |

(10.8)

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Fig. 10.17 Lyapunov exponent estimation

where . j is defined as follows: j ∑ .

i=1

λi > 0 and

j+1 ∑

λi < 0.

(10.9)

i=1

Acknowledgements The author would also like to acknowledge the support of the Brazilian Research Agencies CNPq, CAPES and FAPERJ. The help of Guilherme V. Rodrigues with the Figures is also acknowledged.

References 1. Alligood, K., Sauer, T., Yorke, J.A.: Chaos : an Introduction to Dynamical Systems. Springer Berlin Heidelberg Imprint Springer, Berlin, Heidelberg (1997) 2. Briggs, J., Peat, F.D.: Seven life lessons of chaos : timeless wisdom from the science of change. HarperCollinsPublishers, New York (1999) 3. Franca, L., Savi, M.: Evaluating noise sensitivity on the time series determination of lyapunov exponents applied to the nonlinear pendulum. Shock and Vibration 10(1), 37–50 (2003). https:// doi.org/10.1155/2003/437609 4. Gleick, J.: Chaos : making a new science. Minerva, London (1996) 5. Grebogi, C., Ott, E., Pelikan, S., Yorke, J.A.: Strange attractors that are not chaotic. Physica D 13(1–2), 261–268 (1984). https://doi.org/10.1016/0167-2789(84)90282-3 6. Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, New York (1983) 7. Hirsch, M.W., Smale, S., Devaney, R.L.: Differential equations, dynamical systems, and an introduction to chaos. Elsevier/Academic Press, Amsterdam, Boston (2004)

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8. Kantz, H., Schreiber, T.: Nonlinear time series analysis. Cambridge University Press, Cambridge (2003) 9. Kapitaniak, T.: Chaotic Oscillations in Mechanical Systems. Wiley–Blackwell (1991) 10. Lorenz, E.: The essence of chaos. University of Washington Press, Seattle (1995) 11. Lorenz, E.N.: Deterministic nonperiodic flow. Journal of Atmosferic Science 20(2), 130–141 (1963). https://doi.org/10.1175/1520-0469(1963)0202.0.co;2 12. Moon, F.C.: Chaotic and fractal dynamics : an introduction for applied scientists and engineers. Wiley, New York (1992) 13. Mullin, T.: The Nature of chaos. Clarendon Press Oxford University Press, Oxford, New York (1993) 14. Nayfeh, A.: Nonlinear oscillations. Wiley, New York (1979) 15. Netto, D.M.B., Brandão, A., Paiva, A., Pacheco, P.M.C.L., Savi, M.A.: Estimating lyapunov spectrum on shape-memory alloy oscillators considering cloned dynamics and tangent map methods. Journal of the Brazilian Society of Mechanical Sciences and Engineering 42(9) (2020). https://doi.org/10.1007/s40430-020-02553-6 16. Ott, E.: Chaos in dynamical systems. Cambridge University Press, Cambridge, U.K. New York (2002) 17. Parker, T.S., Chua, L.O.: Practical numerical algorithms for chaotic systems. Springer Verlag, Berlin, New York (1989) 18. Savi, M.A.: Rhythms of Nature. Editora E-papers (in Portuguese) (2014) 19. Savi, M.A.: Nonlinear Dynamics and Chaos. Editora E-papers (in Portuguese) (2017) 20. Savi, M.A., Pacheco, P.M.C.L.: Chaos and hyperchaos in shape memory systems. International Journal of Bifurcation and Chaos 12(03), 645–657 (2002). https://doi.org/10.1142/ s0218127402004607 21. Soriano, D.C., Fazanaro, F.I., Suyama, R., de Oliveira, J.R., Attux, R., Madrid, M.K.: A method for lyapunov spectrum estimation using cloned dynamics and its application to the discontinuously-excited FitzHugh–nagumo model. Nonlinear Dynamics 67(1), 413–424 (2011). https://doi.org/10.1007/s11071-011-9989-2 22. Stewart, I.: Does God Play Dice. Jorge Zahar Editor (in Portuguese) (1991) 23. Strogatz, S.: Nonlinear dynamics and chaos : with applications to physics, biology, chemistry, and engineering. Westview Press, Cambridge, MA (2000) 24. Thompson, J.M.T., Stewart, H.B.: Nonlinear dynamics and chaos : geometrical methods for engineers and scientists. Wiley, Chichester, West Sussex, New York (1986) 25. Wiggins, S.: Introduction to applied nonlinear dynamical systems and chaos. Springer-Verlag, New York (1990) 26. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining lyapunov exponents from a time series. Physica D 16(3), 285–317 (1985). https://doi.org/10.1016/0167-2789(85)900119

Chapter 11

Dynamical Integrity and Its Background Stefano Lenci

Abstract This chapter summarizes the main elements of dynamical integrity, the branch of the dynamical systems theory that is aimed at checking the robustness, or practical stability, of solutions of evolutionary systems governed by differential equations (continuous time) or maps (discrete time). Although developed in the field of engineering, the ideas and results apply to any dynamical systems, irrespective of the problem they are aimed to describe. First, the background is illustrated, and the main aspects of dynamical systems are summarized. The classical concept of stability is then recalled, to understand the starting point of dynamical integrity and how this is aimed at adding new and relevant information with respect to classical knowledge. The main elements of a robustness analysis, namely the safe basin, the integrity measure and the integrity profiles, are finally reviewed, and the regularity of the integrity profiles is studied for the first time.

11.1 Introduction In the pre-scientific era, when all was based on observations, the accepted viewpoint was that if a state of a given mechanical system cannot be seen, it simply does not exist. All was “rule of thumb” based on several previous observations and, mainly, on a trial and error (failure!) approach. We love and are fascinated by the Roman bridges that still exist and still carry loads, but sometimes we forget that to reach that result many bridges collapsed before, and many collapses were needed to choose the right shape and dimensions. The designs were mainly based on small changes of previous structures that were observed to be adequate. This was the adopted methodology for ages, and allowed in any case the building of wonderful structures, not only in civil but also in mechanical engineering (refer for example to the drawings of Leonardo da Vinci). S. Lenci (B) Polytechnic University of Marche, via Brecce Bianche 12, Ancona, Italy e-mail: [email protected]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. R. Castilho Piqueira et al. (eds.), Lectures on Nonlinear Dynamics, Understanding Complex Systems, https://doi.org/10.1007/978-3-031-45101-0_11

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A breakthrough happened with the advent of the scientific era, due to Galileo [1], notably by his famous “Discorsi...” [2], and other giants of that age (without however forgetting that the roots go back to ancient Greeks, in particular to Aristotele [3]). The principal heritage was the so-called scientific method, based on the reproducibility of the results by independent scholars. The other major step was the introduction of mathematics and modelling as a fundamental tools, above all in science (notably in physics) and engineering. Galileo said “The book of nature is written in the language of mathematics”. Cutting edge contributions have been made by Newton [4], among many others, of course. The relationship between mathematics and mechanics/physics has been deeply investigated [5–8], and significantly contributed to the development of science and engineering and, as a by product, to technological development. It produced a huge amount of results, among which the theory of elasticity for continuous bodies, in the static field, and Newton law and Lagrangian theory, in the dynamical regime, are among the most important results, limiting to the so-called “classical mechanics”. Even nowadays, many scientific journals are named “applied mathematics and mechanics”, or similar names, to testify that this strong connection still continues— and is still very fruitful. One of the main curiosity of the application of mathematics to mechanics is that it allows us to predict the existence of solutions (in particular equilibrium and periodic orbits) that cannot be seen in the real world. Accepting that something you cannot see exists (in mechanical engineering) was hard, and in fact this puzzled people for a long time. A lot of efforts were needed to understand that some kind of “quality” of the solutions should be introduced, in addition to its (theoretically predicted) existence. This is what nowadays is known as stability, which was notably introduced in a rigorous way by Lyapunov in 1892 [9], who recognized the inspiration of Poincaré [10]. After this seminal work, a huge amount of research has been done on the topic of stability [11–13], including, e.g., follower forces [14], stochastic systems [15] and non-smooth systems [16], and much knowledge has been gained. The latter reference contains also a short but very interesting history of stability, which is referred to for further deepening. From a theoretical point of view one of the most important developments was the local bifurcation theory [17–21], which systematizes and generalizes results that often were already been obtained by engineers (for example the well-known Euler buckling load for slender compressed beams [22], which only much later was recognized as a pitchfork bifurcation). The discovery that different mechanical problems may undergo the same bifurcation, and thus the same phenomenon, was a very important breakthrough that, among others, permitted the exploitation of some analogies. One of the major results was that there are very few local generic bifurcations, thus simplifying the investigations. Stability was addressed also from an experimental point of view [23–26]. Here the scientific community faced with another big problem, namely not being able, in certain cases, to reproduce experimentally the critical load (or, more generally, the bifurcation point) theoretically predicted, even beyond the obvious approximations

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due to experimental errors. Alternatively, but with a similar meaning, structures— mainly cylindrical and spherical shells—built according to theoretical results collapsed catastrophically [27]. Again, this was a challenging issue, and was solved mainly by Koiter [28], with successive important contributions, among many others, by Budiansky [29] and Tvergaard [30]. Koiter realized that bifurcation points can be strongly sensitive to imperfections. “Strongly sensitive” means that they change a lot even for small changes of the parameters. And of course the real world is full of imperfections, even if they can be kept small and controlled in refined experiments. But there is no hope of eliminating them at all! Thus, the critical threshold considering the imperfections can be much different from the theoretical one (without considering imperfections), and this fits with experimental observations. Koiter focused on imperfections in the system, i.e. uncertainty in the knowledge of the parameters like geometry (length, thickness, eccentricity of the loads, etc.), boundary conditions, material characteristic (Young modulus, etc.), etc. This was again systematized by mathematicians, and gave rise to the structural stability that, as said, deals with the effects of the imperfections of the system parameters on the outcome [31]. It was notably recognized that, in general, only two bifurcations, named “saddlenode” and “Hopf”, are structurally stable for equilibrium points, and only three are structurally stable for periodic solutions, “saddle-node”, “period-doubling” and “Neimark” [19–21]. Only these are robust and expected to be observed in the real world. Other bifurcations are structurally stable only within subsystems, like for example symmetrical ones, and of course are not expected to occur, strictly speaking, in practice. Who can guarantee perfect symmetry? Imperfections may occur not only in the parameters of the system, but also in initial conditions. And, while imperfections of the parameters are given (even if unknown) and remain the same in time (forgetting about damages and ageing, which are exceptional and require a long time, respectively), imperfections in initial conditions are pervasive and, moreover, can occur at any time, due to any time dependent perturbation. And thus, a check today may not be valid tomorrow, without considering possible changes in initial conditions. But considering imperfections with respect to initial conditions is not stability? Not exactly, because stability deals in an indispensable way with infinitesimal perturbations. For example, studying stability through the eigenvalues of an appropriate matrix, which is the common practice, heavily rests on this fundamental hypothesis. It was Thompson [32], in a series of papers in the nineties of the twentieth century [33–39], who realized that considering only infinitesimal perturbations in initial conditions is not enough, since in the real world imperfections on initial conditions could be larger than infinitesimal, while remaining small, since otherwise they cannot be called imperfections, as they are visible and can be “easily” measured and considered in the analysis. Although it may appear a minor improvement, it is indeed a major advance, as it involves new fundamental aspects. For example, the local analysis based on eigenvalues can be insufficient, and a global analysis is commonly needed [40–43],

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requiring other tools, in particular “basins of attraction” [44–47], that while being conceptually simple can have an extremely complex shape [48–51] and, on top of that, are very difficult to be determined when the dimension of the system is larger than, say, two mechanical degrees of freedom (i.e. the phase space is 4D) [42, 52–54]. This third cutting edge was later named dynamical integrity [55, 56], and it has been deeply investigated in recent years [43, 57], including an historical perspective [58]. Many studies have been done addressing the main theoretical issues [56, 59, 60], including the definition of safe basins and the introduction of appropriate integrity measures [59, 61, 62]. It has been applied to the Duffing [63, 64] and Helmholtz [65] oscillators, pendulum [66], Augusti model [67, 68], guyed mast [69], micromechanics [70–75], cylindrical shells [76, 77], carbon nanotubes [78] and other mechanical systems [79–82]. Also the stochastic aspects have been studied [83–88], as well as control of erosion [56, 57, 59, 63, 65, 70, 89]. Notably, the ability of dynamical integrity in understanding apparently unexpected experimental results has been underlined [90], both on the micro [91, 92] and macro [93] scale. Last, but not least, the relevance of dynamical integrity for safe engineering design has been also highlighted [43, 94, 95]. Although dynamical integrity dates back to 30 years ago, its relevance is not yet understood and realized by the engineering community, not even by scholars. In fact, it is not yet entered in the background of engineers, neither standard nor specialized, and this lack of knowledge has the consequence that we are missing a lot of opportunities. In addition to the common resistance, delay, and fear that accompany any new important discovery, it is felt that this is also due to the fact that it may require a tool, the basins of attraction (or its generalization “safe basin” [56, 59]), that could be difficult to determine and requires a certain expertise. And even not using this tool, and accepting a reduced level of knowledge and still catching the phenomenon of small but not infinitesimal perturbations, it is in any case expensive from a computational point of view. To underline the importance of dynamical integrity, it is sufficient to recall that saddle-nodes are structurally stable, so are not sensitive, and thus are expected to be observable in the real world. But sometimes they are not, even in refined experiments. This is because commonly the basin of attraction shrinks when approaching the bifurcation point, and this clearly illustrates, by dynamical integrity arguments, why it is not possible to reach bifurcations points in the real world, where imperfections in initial conditions are pervasive and easily lead to another solution outside the basin, when the basin has a too small compact part. The goal of this work is to better illustrate the main steps of this story, that is told from the viewpoint of the author, and thus is not expected to be exhaustive nor fully objective (although I tried my best). What a dynamical system is, which is a prerequisite for the following parts, constitutes the subject of Sect. 11.2. Then, starting from stability (Sect. 11.3)—or from the “past”—I will arrive at the dynamical integrity (Sect. 11.4)—or the “future”—, also addressing some technical points that are needed for full understanding. The paper ends with some conclusions and challenges for the future (Sect. 11.5).

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11.2 Dynamical Systems 11.2.1 A Continuous Time Dynamical System A continuous time dynamical system is basically described by its time-differential governing equation ˙ (t) = f(x(t), t; µ), .x (11.1) together with the initial condition x(t0 ) = x0 ,

(11.2)

f(x, t; µ) : Ω × I × Ξ → Ω,

(11.3)

.

where .

is the function that practically defines the system. Obtaining (11.1) from the real (physical, engineering, economics, biological, chemical, etc.) system is often a difficult work that is known as “modelling” and requires the capability to mathematically describe the underlying phenomena. I assume that this work has been previously done, and our starting point is (11.1) (or forthcoming (11.6)). In (11.1).x(t) = x(t; t0 , x0 , µ) ∈ Ω is the state variable (the unknown of the problem), .Ω ∈ H is the space of configurations, which is a subset of a Hilbert space .H (even a Banach or metric space could be fine, but not less, since we need a distance in the space of configurations to introduce stability later on), overdot means derivative with respect to the time .t ∈ I variable, . I ∈ R is the time interval, .Ξ ∈ R M is the space of parameters, . M the number of parameters (e.g. Young modulus, length, stiffness, etc.), .t0 the initial time and .x0 the initial condition. . I × Ω, where the initial conditions .(t0 , x0 ) lie, is the phase space. If .H = R N , i.e. if .x is . N -vector, the system is . N -dimensional and the governing equations are a system of . N ordinary differential equations (ODEs). If .H is a functional space, i.e. if .x is a function (of other variables, for example space variables), the system is infinito-dimensional and the governing equations are partial differential equations (PDEs) or delay equations. To focus on the main ideas without excessive mathematical difficulties, I limit this work to finite dimensional systems, and with not so large values of . N , indeed. There exist reduction techniques that permit approximating large or infinite-dimensional systems with low dimensional ones [96]. Furthermore, I consider only smooth systems, i.e. .f is sufficiently differentiable with respect to .t. For non-smooth systems the reader is referred to [16, 97–99]. For time delay systems, those for which .f(x, t − τ ; µ), .τ > 0 being the delay, I quote [100, 101]. For time varying systems, where the dimension . N changes in time, I refer to [102, 103]: they are often related to non-smooth systems.

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Second order derivatives .x¨ can be transformed to first order by introducing the velocity.y = x˙ , so that.x¨ = y˙ . For example, the second order system.x¨ = f(x, x˙ , t; µ) becomes ( ) ( ) x˙ y = , . (11.4) y˙ f(x, y, t; µ) which is first order. The “price” is that the number of variables doubles. Higher order derivatives can be reduced to first order in a similar way. When .f does not depend on .t, i.e. .x˙ = f(x; µ), the system is called autonomous. Any non-autonomous system can be transformed in an autonomous one by considering the time as an extra variable: ( ) ( ) x˙ f(x, θ; µ) = . . 1 θ˙

(11.5)

I further assume that the solution .x(t) = x(t; t0 , x0 , µ): 1. exists and is unique for every.t ≥ t0 , i.e.. I = [t0 , ∞[, so that the dynamical system always evolves forward in time; 2. continuously depends on the initial conditions .(t0 , x0 ); 3. continuously depends on the parameters .µ. This can be proved under mild conditions on .f, i.e. that it is Lipschitz continuous. To simplify the notation, in the following part of the chapter the dependence of .f on .µ is omitted when not explicitly used.

11.2.2 A Discrete Time Dynamical System A discrete time dynamical system is basically described by the recurrence relation x

. n+1

= g(xn , n; µ),

(11.6)

together with the initial condition .x0 (I implicitly assume, without loss of generality, that the system starts at .n = 0). The function g(x, n; µ) : Ω × I × Ξ → Ω,

.

(11.7)

also called map, defines the system. Discrete systems naturally occurs in applications where relevant information are available only at fixed time intervals (for example, in economics, the Gross Domestic Product (GDP) that is computed every year). In (11.6) .xn (x0 , µ) ∈ Ω is the state variable, .Ω ∈ H is the phase space, which is a subset of a Hilbert space .H, .n ∈ I is the discrete time variable, . I ∈ Z is the time interval, .Ξ ∈ R M is the space of parameters and . M the number of parameters.

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If .H = R N , i.e. if .x is . N -vector, the system is . N -dimensional. If .H is a functional space, i.e. if .x is a function (of other variables, for example space variables), the system is infinito-dimensional. When .g does not depend on .n, i.e. .xn+1 = g(xn ; µ), the system is called autonomous. Any non autonomous system can be transformed into an autonomous one by considering the time as an extra variable, as done for continuous systems. Since .g maps .Ω in itself, the solution is always definite forward in time (i.e. .n ≥ 0): x = g(x0 , 0),

. 1

x2 = g(x1 , 1) = g(g(x0 ), 1) = g2 (x0 ), ... ,

(11.8)

xn = gn (x0 ).

(11.9)

We can consider also backward iterations, which however could be not unique (because the inverse .g−1 could be not unique) or not defined (because .g could be not surjective). To simplify the notation, in the following the dependence of .g on .µ is omitted when not explicitly used.

11.2.3 Poincaré Section and Poincaré Map Poincaré [104] proposed a clever and useful tool that permits transforming a continuous dynamical system into a discrete one. A Poincaré section is a. N − 1 subset (hyper-surface).Σ in the. N dimensional state space of a continuous time system. In special cases it can be also a lower dimensional subset, even if this is not common..Σ does not need to be planar or regular, but usually it is. Instead, it is required to be transverse to the vector field: n · f(x, t) /= 0,

.

(11.10)

where .n is the unit vector perpendicular to .Σ (Fig. 11.1a). The Poincaré section is not unique, and can be properly chosen to simplify the analysis. For periodic systems, those for which f(x, t) = f(x, t + P)

.

(11.11)

for a certain (minimal) . P > 0, .Σ is taken perpendicular to the time variable and is named “stroboscopic” section (Fig. 11.1b). In this case the condition (11.10) is automatically satisfied since.n · f(x, t) = 1 because.n = (0, 0, . . . , 0, 0, 1)T (see (11.5)). A Poincaré section is useful if the initial conditions starting on .Σ (or at least a subset of it) come back to .Σ after a certain time: . A ∈ Σ goes to . B ∈ Σ in Fig. 11.1.

308

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caré Poin

f(x,t) A

n ectio ré s

ion sect

A

x(t=0)

forward integration in time

n

B

B

on secti caré Poin (a)

ca Poin

x(t=P)

time

(b)

0

P (period)

Fig. 11.1 Poincaré section .Σ and Poincaré map. a Generic case and b stroboscopic

The return time can vary from point to point. In this case, the Poincaré map is defined as the function that associates any (starting) point in .Σ to its returning point (also belonging to .Σ): it is the function .g : A → B in Fig. 11.1. Commonly the returning point is the first one, but this is not necessary, and a subsequent return point can be considered as well. For example, it may be required to be the first returning point entering from the same side, i.e. .sign(n · f(x, t))| A = sign(n · f(x, t))| B . For the stroboscopic section, the naturally defined Poincaré map is obtained by exploiting the periodicity of the system (Fig. 11.1b). In this case the returning time is fixed and equal to . P. The Poincaré map transforms the original continuous system in a discrete one, where . A = xn and . B = xn+1 . In general its expression cannot be found in closed analytical form, but it can be easily obtained by numerically integrating (11.1) or with other approximation techniques. When the map .g is obtained by numerical methods or analytical approximations, the evaluation of the gradient matrix .∇g (to be later used for stability issues) is not straightforward and needs extra care.

11.2.4 Different Kinds of Motion There are basically five different kinds of solution of (11.1) and (11.6): 1. 2. 3. 4. 5.

equilibrium (continuous systems) or fixed (discrete systems) points; periodic; quasi-periodic; homoclinic or heteroclinic; chaos.

They will be discussed in the following subsections.

11 Dynamical Integrity and Its Background

11.2.4.1

309

Equilibrium and Fixed Points

An equilibrium position is a solution of (11.1) that does not depend on (continuous) time, .x(t) = xe . (11.12) It is clearly a solution of the algebraic (if .H = R N ) or spatial PDE (if .H is a functional space) equation .f(xe , t) = 0, (11.13) and requires that .f does not depend on .t in .xe . It corresponds to the system at rest, and for this reason it is a “static” solution. A fixed point is a solution of (11.6) that does not depend on (discrete) time, x = xf.

. n

(11.14)

It is clearly a solution of the algebraic (if .H = R N ) or spatial PDE (if .H is a functional space) equation .x f = g(x f , n), (11.15) and requires that .g does not depend on .n in .x f . Equation (11.15) can alternatively be written as f(x f , n) = x f − g(x f , n) = 0,

.

(11.16)

and thus we see that finding fixed points of maps is mathematically equivalent to finding equilibrium positions of continuous systems. Equilibrium and fixed points are the simplest solution of any dynamical system.

11.2.4.2

Period Solutions

A periodic solution of time continuous systems is a solution of (11.1) such that x(t + T ) = x(t),

.

(11.17)

where .T > 0 is the smallest number satisfying (11.17) and is named “period”. . f = 1/T is the “frequency” and .ω = 2π/T is the “circular frequency”. Periodic solutions commonly occur when .f is periodic, see (11.11) and Fig. 11.2, and has the same period, .T = P, or an integer multiple of that, .T = n P, .n ∈ N (in this latter case the solution is named .n-periodic). However, they can also have a different period, .T /= n P, and .T and . P can be even incommensurable. Actually, the periodicity of .f is not required. A system can have a periodic solution even if .f is independent of .t (Fig. 11.3).

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S. Lenci

(b)

Fig. 11.2 Periodic solution of the equation .x¨ + 0.2 x˙ + x + x 3 = 0.3 sin(5t). a Time history; b phase portrait. The blue dot is the point in the Poincaré section

(a)

(b)

Fig. 11.3 Periodic solution of the equation .x¨ + x + x 3 = 0. a Time history; b phase portrait

Let us consider the point .x(0) of a periodic solution of a continuous time system, and consider a . N − 1 hyperplane .Σ passing through .x(0) and not parallel to the vector field, according to (11.10). At least locally, by the continuity of the solution with respect to the initial conditions, it is a Poincaré section. By (11.17), when .t = T we are back on .Σ, and this allow the construction of a Poincaré map. But not only that, we are back exactly on the initial point (i.e. . A = B in Fig. 11.1a), and this exactly corresponds to the definition (11.15) of a fixed point of the Poincaré map. The remarkable conclusion is that any periodic solution of a continuous time system corresponds to a fixed point of an associated discrete system (see Sect. 11.2.4.1), which actually is not uniquely defined (since .Σ is not unique). This fixed point is just a point on the Poincaré section (it is the blue dot of Fig. 11.2b).

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Similarly to what has been done for continuous time systems, a periodic solution can be defined also for discrete systems. It is a solution of (11.6) such that x

. n+H

= xn ,

(11.18)

where. H > 0 is the smallest integer satisfying (11.18) and, again, is named (discrete) “period”. Note that (11.18) is equivalent to x = g H (x0 ),

(11.19)

f(x0 ) = x0 − g H (x0 ) = 0,

(11.20)

. 0

and to .

i.e. mathematically it is equivalent to finding a fixed point of the . H -iterated map.

11.2.4.3

Quasi-periodic Solutions

A rigorous definition of a quasi-periodic solution is difficult, and I can use the Wiggins intuitive definition: “a quasi-periodic orbit is just a two-frequency solution of the ordinary differential equation, where the two frequencies are incommensurate” [20]. It easily extends to many (incommensurate) frequencies, and, accordingly, it can be identified by the Fourier transform, when there is more than one peak corresponding to different, incommensurate, frequencies (Fig. 11.4d). The fact that the involved frequencies are incommensurate is crucial, otherwise the solution would be periodic, maybe with a very large period. Instead, a quasi-periodic solution never repeats in time. In the phase space a quasi-periodic solution (with two incommensurate frequencies) fulfills a torus (Fig. 11.4c), with one period for running along the axis torus and the other one for winding around the axis. Taking a Poincaré section transversal to the torus axis, we see that a quasi-periodic solution is described by a close curve on the section (Fig. 11.4b). This could be another way to recognize quasi-periodic solutions, even if it works only for two incommensurate frequencies.

11.2.4.4

Homoclinic and Heteroclinic Solutions

The homoclinic solution .x h (t) of a continuous time dynamical system approaches another solution .xs (t) (commonly an equilibrium or a period solution) forward and backward in time (Fig. 11.5): .

lim x h (t) = xs (t).

t→±∞

(11.21)

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(a)

(b)

(c)

(d)

√ Fig. 11.4 Quasi-periodic solution of the equation.x¨ + 0.001x˙ + 4(1 + 0.1 cos 3t)x + x 2 + x 3 = 1.5 cos(t) [105]. a) Time history; b) phase portrait. The blue closed curve is the trace on the Poincaré section; c) the torus in a 3D representation; d) modulus of √ the Fourier transform. The main peaks are at .ω1 = 2ω3 − ω4 , .ω2 = ω4 − ω3 , .ω3 = 1 (red), .ω4 = 3 (green), .ω5 = 2ω3 , .ω6 = ω3 + ω4 , .ω7 = 3ω3 , .ω3 and .ω4 being incommensurate

The heteroclinic solution .x h (t) approaches one solution .xs1 (t) (commonly an equilibrium or a period solution) forward in time and a different solution .xs2 (t) (commonly an equilibrium or a period solution) backward in time (Fig. 11.6): .

lim x h (t) = xs1 (t),

t→∞

lim x h (t) = xs2 (t).

t→−∞

(11.22)

Homoclinic and heteroclinic solutions .x h,n of discrete time systems are defined in a similar way (it is sufficient to exchange .limt→±∞ with .limn→±∞ ).

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(b)

(a)

Fig. 11.5 Homoclinic solution.x(t) = b phase portrait

√

2/ cosh t of the equation.x¨ − x + x 3 = 0. a Time history;

(a)

(b) √ Fig. 11.6 Heteroclinic solution .x(t) = tanh (t/ 2) of the equation .x¨ + x − x 3 = 0. a Time history; b phase portrait

11.2.4.5

Chaotic Solutions

Chaotic solutions are the most recently discovered, and the birth of the modern chaos theory is commonly credited to the celebrated work of Edward Lorenz [106], although Poincaré previously provided deep understanding. Although fascinating, a deep description of chaos theory, which is far from trivial from a mathematical point of view, is out of the scope of the present chapter, and the interested reader can refer to [19, 20, 107–111]. A chaotic trajectory (Fig. 11.7a) is a solution that never repeats in time (so it is not periodic) and has infinitely many incommensurate frequencies (so it is not quasiperiodic and its Fourier spectrum is somehow continuous), and does not approach any

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(b)

Fig. 11.7 Chaotic solution (attractor) of the equation .x¨ + 0.15x˙ − x + x 3 = 0.3 sin(t). a Time history; b Stroboscopic Poincaré section

other solution forward or backward in time (so it is not homoclinic or heteroclinic). Although it never repeats, forward in time (and maybe after a very long interval of time) it gets back arbitrarily close to any of its points. However, in short time intervals points diverge from each other. For continuous time systems, in an appropriate Poincaré section, a chaotic solution is represented by infinitely many dots, which however are organized along certain geometrical structures (this is the difference with respect to stochastic systems, whose trace on a Poincaré is given by a disorganized cloud of points), as shown in Fig. 11.7b. Instead of a single, generic chaotic solution, commonly one is more interested in chaotic attractors, i.e. chaotic solutions that are the limits of some neighborhood solutions, or in chaotic behaviours, which are characterized (at least) by the following properties. • Sensitivity to initial conditions. This means that initial conditions arbitrarily close will diverge in a sufficiently large time. This clearly leads to unpredictability in a deterministic system (different from unpredictability in a stochastic system—this is crucial), and this is the most famous property of chaos, often named “butterfly effect” after Lorenz (see [109], Chap. 7). It does not violate the continuity with respect to initial conditions, which is obtained in fixed time. It is measured by (the positiveness) of the maximum Lyapunov exponent. • Ergodicity or mixing. Any given region or open set will eventually overlap with any other given region. It is related to the existence of a dense orbit: there is at least one orbit which passes in time as close as one wishes to any point of the attractor, and it is the property that shows how the orbits mix in the phase space, corresponding to a rough notion of erratic behaviour. • Density of periodic orbits. To any trajectory of the system there exists a periodic solution that stays close to it for a long time. It means that a chaotic attractor may

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appear as an erratic collection of apparently periodic solutions. It is the less common and less intuitive, and apparently the one with fewer practical consequences. The theoretical importance of periodic orbits is underlined by the famous paper by Li and Yorke [112], which is a particular case of a more general result obtained by Sharkovskii [113], and which apparently introduced the word “chaos”.

11.3 Stability Stability investigates an important quality of solutions of any dynamical system, namely whether starting from a slightly different initial condition, the solution will remain close (stable) or diverge (unstable) forward in time. Stability is a fundamental property, since unstable solutions cannot be observed in the real world, which is full of perturbations and imperfections that lead to small variations of initial conditions.

11.3.1 Stability of Equilibrium Points I start with the various definitions of the Lyapunov stability of an equilibrium point x of a continuous time system. I assume that it is autonomous: because of (11.5), this is not restrictive.

. e

1. .xe is stable if, for every .ε > 0 (sufficiently small), there exists a .δ(ε, t0 ) > 0 such that .d(x(t), xe ) < ε, .∀t > t0 , for every initial condition .x(t0 ) such that .d(x(t0 ), xe ) < δ(ε, t0 ), where .d is the distance in the space of configurations .Ω. 2. .xe is uniform stable if, in the definition (1), .δ does not depend on .t0 . 3. .xe is asymptotically stable if there exists a .δ(t0 ) > 0 such that .limt→+∞ d(x(t), xe ) = 0 for every initial condition .x(t0 ) such that .d(x(t0 ), xe ) < δ(t0 ). 4. .xe is uniform asymptotically stable if, in the definition (3), .δ does not depend on .t0 . 5. .xe is exponentially stable if there exist .δ(t0 ) > 0, .b > 0 and .c > 0 such that −c(t−t0 ) .d(x(t), xe ) < be d(x(t0 ), xe ) for every initial condition .x(t0 ) such that .d(x(t0 ), xe ) < δ(t0 ). 6. .xe is uniform exponentially stable if, in the definition (5), .δ does not depend on .t0 . It is obvious that if .xe is exponentially stable it is also asymptotically stable, and if it is asymptotically stable it is also stable. It is worth remarking that, because of the request .d(x(t0 ), xe ) < δ, stability is a local property, in a neighborhood of the equilibrium point .xe . Global stability is obtained if the previous properties hold for every initial condition in the space of configurations. While for linear systems global stability is the common case, for

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nonlinear systems it is quite uncommon, and multistability (i.e. the simultaneous occurrence of different stable solutions) may occur.

11.3.2 Stability of Fixed Points I report here the various definitions of the Lyapunov stability of an fixed point .x f of a discrete time system. I assume that it is autonomous, which is not restrictive. 1. .x f is stable if, for every.ε > 0 (sufficiently small), there exists a.δ(ε) > 0 such that .d(xn , x f ) < ε, .∀n > 0, for every initial condition .x0 such that .d(x0 , x f ) < δ(ε), where .d is the distance in the space of configurations .Ω. 2. .x f is asymptotically stable if there exists a.δ > 0 such that.limn→+∞ d(xn , x f ) = 0 for every initial condition .x0 such that .d(x0 , x f ) < δ. 3. .x f is exponentially stable if there exist .δ > 0, .b > 0 and .c > 0 such that −cn .d(xn , xe ) < be d(x0 , x f ) for every initial condition .x0 such that .d(x0 , x f ) < δ. Again, these are local definitions of stability, which hold only in neighborhood of xf. It is worth underlining that, thanks to an appropriate Poincaré map introduced in Sect. 11.2.4.2, the stability of fixed points of discrete time systems can be immediately used to define the stability of periodic solutions of continuous time systems, which is therefore not discussed explicitly.

.

11.3.3 Stability of Generic Solutions Let us try to extend the definition of Lyapunov stability to any reference solution x (t) of a continuous time system.

. r

1. .xr (t) is stable if, for every .ε > 0 (sufficiently small), there exists a .δ(ε, t0 ) > 0 such that .d(x(t), xr (t)) < ε, .∀t > t0 , for every initial condition .x(t0 ) such that .d(x(t0 ), x f (t0 )) < δ(ε, t0 ), where .d is the distance in the space of configurations .Ω. I do not report the stronger definitions of stability shown in Sect. 11.3.1, which can be easily adapted. I address, instead, a limitation of the previous definition, namely that the closeness between .x(t) and .xr (t) has to be checked at the same time .t. Sometimes this could be too restrictive, and it could be enough that the orbits in the phase space remain close, without the specific configurations (at each instant in time) being close. This leads to the following definition. 2. .xr (t) is orbitally stable if, for every .ε > 0 (sufficiently small), there exists a ' ' .δ(ε, t0 ) > 0 and a .t (t, t0 ) such that .d(x(t (t, t0 )), xr (t)) < ε, .∀t > t0 , for every

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initial condition.x(t0 ) such that.d(x(t0 ), x f (t0 )) < δ(ε, t0 ), where.d is the distance in the space of configurations .Ω. 3. .xr (t) is uniform orbitally stable if, in the definition (2), neither .t ' and .δ depends on .t0 . To extend the previous definitions to generic solutions of discrete time systems is straightforward, and is not explicitly reported to limit the length of the work.

11.3.4 Local Analysis Around an Equilibrium Point In a neighborhood of an equilibrium point .xe of a continuous time system we can introduce the small variable .y(t) = x(t) − xe and expand the vector field in a Taylor series with respect to .y(t), obtaining: y˙ (t) = ∇f(xe )y(t) + . . . ,

.

(11.23)

where .∇f(xe ) is the gradient of the vector field computed at .xe . The general solution of (11.23) is given by y(t) = Σi αi eλi t yi ,

.

(11.24)

where .λi are the complex eigenvalues and .yi the eigenvectors of the linear operator ∇f(xe ), which could be a PDE if the system is infinito-dimensional. The .αi depend on the initial conditions and are not relevant here. Remembering the Euler formula .eλi t = eRe(λi )t [cos Im(λi )t + I sin Im(λi )t], we observe that .Re(λi ) determines the amplitude of the motion and .Im(λi ) determines the oscillating behavior. .xe is said hyperbolic if all the eigenvalues have a real part different from zero, .Re(λi ) / = 0; .∀i, non-hyperbolic if at least one eigenvalues has a zero real part, .∃ j such that .Re(λ j ) = 0. For hyperbolic equilibrium positions the stability is fully determined by the eigenvalues, and is a simple consequence of (11.24): .

• .xe is exponentially stable if all eigenvalues have a negative real part, .Re(λi ) < 0, .∀i; • .xe is unstable if at least one eigenvalue has a positive real part, .∃ j such that .Re(λ j ) > 0. This is known as the first (or indirect) Lyapunov method. Examples are reported in Fig. 11.8. For non-hyperbolic equilibrium positions the analysis of the eigenvalues is not sufficient to determine the stability (unless one eigenvalue has .Re(λ j ) > 0 so that the system is unstable). This is a consequence of the fact that in the direction of the eigenvector .y j with .Re(λ j ) = 0 the nonlinear terms (neglected in the linear analysis)

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Fig. 11.8 a Exponentially stable hyperbolic; b unstable hyperbolic; c non-hyperbolic

Fig. 11.9 a Stable node; b unstable node; c sink or stable focus; d source, repellor or unstable focus; e saddle f center equilibrium point

(a)

(b)

(c)

(a)

(b)

(c)

(d)

(e)

(f)

are dominant, because of the absence of the linear term. In this case the stability is determined by higher order terms in the Taylor expansion. The following classification is useful. An equilibrium positions is a: • stable node if all eigenvalues are real and negative (Fig. 11.9a); • unstable node if all eigenvalues are real and positive (Fig. 11.9b); • sink or stable focus if all eigenvalues are imaginary and have a negative real part (Fig. 11.9c); • source or repellor or unstable focus if all eigenvalues are imaginary and have a positive real part (Fig. 11.9d); • saddle if some eigenvalues have a negative real part and other have a positive real part (unstable) (Fig. 11.9e); • center if all eigenvalues have a zero real part (non-hyperbolic equilibrium point – stability undetermined, requires nonlinear terms) (Fig. 11.9f).

11.3.5 Local Analysis Around a Fixed Point Similarly to what has been done for continuous systems in Sect. 11.3.4 I study the local stability of the fixed point .x f of a discrete time system (and thus of periodic orbits of continuous time systems) by introducing the small variable .yn = xn − x f and by expanding the vector field in a Taylor series with respect to .yn , obtaining: y

. n+1

= ∇g(x f )yn + . . . .

(11.25)

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319

Let .ρi and .yi be the eigenvalues (possibly complex – conjugate) and (normalized) eigenvectors of the linear operator .∇g(x f ), namely ρ yi = ∇g(x f )yi .

(11.26)

. i

Apart from pathological cases, .yi is a base of .Ω, i.e. every vector .y0 (initial condition, or starting point) can be written as y = Σi αi yi .

(11.27)

. 0

It follows that y = ∇g(x f )y0 = ∇g(x f )Σi α j yi = Σi αi ∇g(x f )yi = Σi αi ρi yi

(11.28)

y = ∇g(x f )y2 = ∇g(x f )Σi αi ρi yi = Σi αi ρi ∇g(x f )yi = Σi αi ρi2 yi .

(11.29)

. 1

and . 2

Iterating we get y = Σi αi ρin yi .

(11.30)

. n

Remembering that .ρin = |ρi |n [cos(nθi ) + I sin(nθi )], .θi = arctan[Im(ρi )/ Re(ρi )] being the phase angle of the complex number .ρi , we observe that .|ρi | determines the amplitude of the motion and .θi determines the “rotation” around the fixed point. .x f is said to be hyperbolic if all the eigenvalues have a modulus different from one, .|ρi | /= 1, .∀i; non-hyperbolic if at least one eigenvalue has a modulus one, .∃ j such that .|ρ j | = 1. For hyperbolic equilibrium positions the stability is fully determined by the eigenvalues, and is a simple consequence of (11.30): • .x f is stable if all eigenvalues have modulus less than one, .|ρi | < 1, .∀i; • .xe is unstable if at least one eigenvalue has modulus greater than one, .∃ j such that .|ρ j | > 1. Examples are reported in Fig. 11.10. As for continuous time systems, also here determining the stability of nonhyperbolic points requires higher order (nonlinear) terms in the Taylor development (11.25).

Fig. 11.10 a Stable hyperbolic; b unstable hyperbolic; c non-hyperbolic (a)

(b)

(c)

320 Fig. 11.11 a Stable node; b unstable node; c sink or stable focus; d source, repellor or unstable focus; e saddle f center fixed points

S. Lenci

(a)

(b)

(c)

(d)

(e)

(f)

The following classification is useful. A fixed point is a: • stable node if all eigenvalues are real and have modulus less than one (Fig. 11.11a); • unstable node if all eigenvalues are real and have modulus greater than one (Fig. 11.11b); • sink or stable focus if all eigenvalues are imaginary and have modulus less than one (Fig. 11.11c); • source or repellor or unstable focus if all eigenvalues are imaginary and have modulus greater than one (Fig. 11.11d); • saddle if the eigenvalues are real, some have modulus greater than one and the other have modulus less than one (unstable) (Fig. 11.11e); • center if all eigenvalues have modulus equal to one (non-hyperbolic fixed point – stability undetermined, requires nonlinear terms) (Fig. 11.11f).

11.4 Dynamical Integrity In the Lyapunov definitions of stability reported in Sect. 11.3 it is crucial the fact that the requested properties must be satisfied only for every initial condition .x(t0 ) (for discrete time systems .x0 ) such that .d(x(t0 ), xr (t0 )) < δ (for discrete time systems −100 .d(x0 , xr,0 ) < δ). No matter on how small .δ is! It is sufficient that it exists, .10 is fine. And staying very close to the equilibrium or fixed points is also at the base of the linearized analyses of Sects. 11.3.4 and 11.3.5. While being mathematically consistent, this requirement is not enough in certain practical applications. In fact, if .δ is too small, even imperceptible changes of initial conditions will lead far from the solution. In this case the solution is “practically unstable”, or not “robust”, in spite of being mathematically stable. This concept is illustrated in Fig. 11.12. Both Fig. 11.12a and Fig. 11.12b report stable periodic solutions, but while the left case is also practically stable (since .δ is large enough), the right case is practically unstable and thus, notably, not useful for applications. Dynamical integrity is the branch of dynamical systems theory aimed at investigating these aspects, and determining when a stable (in the Lyapunov sense) solution

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1

1

0

0

-1 -1.8

0

(a)

1.8

-1 -1.8

0

1.8

(b)

Fig. 11.12 Basins of attraction of the stroboscopic Poincaré map of (11.33). The dot represents the periodic solution, and the circle of radius the largest .δ for stability is reported. a . A = 0.01, Lyapunov exponentially stable and practically stable case; b . A = 0.16, Lyapunov exponentially stable but practically unstable

can be utilized in practice (i.e. when it is also practically stable or robust), where there are always small (but not infinitesimal) changes in initial conditions due to imperfections or perturbations, even if the system itself is totally deterministic. A fundamental tool for dynamical integrity is the basin of attraction of a given asymptotically stable solution (also named “attractor”). It is the set of all initial conditions approaching the attractor forward in time. This definition applies for both continuous and discrete time systems. Examples are reported in Fig. 11.12, where different colors are used for different basins. From the view point of basins of attraction, the practical stability, or robustness, entails checking whether the basin of attraction is “large enough”. This is a very delicate issue. In fact, the most intuitive approach would be requiring that the basin of attraction has a large magnitude (their areas in Fig. 11.12). But, unfortunately, often this is not correct. For example, in the case of Fig. 11.12b both red and green basins have about half of the area of the considered region, just as those in Fig. 11.12a. But evidently the associated attractors are not robust, and this is due to the fractality of the basins, which strongly complicates the analysis and requires further understanding and developments, which are illustrated in the following.

11.4.1 Safe Basins The basins of attraction are commonly utilized as a tool for determining the robustness of a given attractor. However, it can be generalized, and the safe basin can be defined

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as the set of initial conditions sharing a given property (whichever), that can be useful for certain scopes. Here is a list of cases, that of course are far from exhaustive: • approaching a given attractor. This gives exactly the basin of attraction; • remaining in a given potential well, irrespective of the final attractor; • involving the transient dynamics, for problems where transient is more important than steady state (impacts, earthquakes, shocks, etc.); • staying very far apart from a dangerous position/attractor (ship capsizing, collapses, etc.); • not overcoming a certain threshold (for example, remaining in the elastic regime); • on the contrary, overcoming a certain threshold (when the goal is to extract a solution from an unwanted position). In spite of the fact that the general definition of safe basins is scope-oriented, and thus most relevant for applications, only basins of attraction have important geometrical properties. In particular, they are bounded by stable manifolds of appropriate saddles [20], and thus can be detected independently [114], a fact that is fundamental from a theoretical point of view for understanding the basin structure, and may also be useful for the basin determination.

11.4.2 Integrity Measures In order to make quantitative evaluations, it is necessary to quantify the robustness of the safe basin by appropriate measures. The simplest measure of the robustness of a safe basin is its magnitude, that is commonly referred to as Global Integrity Measures (GIM). When the basin is not fractal, it is a reliable measure, easy to determine, and more than sufficient for the evaluation purposes. It is the most natural and the first to be introduced [61]. The major drawback of GIM is that it does not take into account the fractal parts, which clearly reduces robustness even if the magnitude of the basin is large (see Fig. 11.12b). In fact, it is only the compact part of the safe basin that contributes to robustness, and thus we need measures that are able to rule out the fractal part and focus on the compact core. This is the most delicate point of the dynamical integrity analysis, and leads to different measures. The Local Integrity Measure (LIM) is the radius of the largest hypersphere (circle in dimension two) centered in the attractor and entirely belonging to the basin. It can be alternatively be seen as the minimum distance between the attractor and its basin boundary [61]. It is evidently effective in ruling out the fractal part. Soliman and Thompson [61] also introduced the Impulsive Integrity Measure (IIM), defined as the minimum distance between the attractor and its basin in the direction of the velocity only. It is clearly relevant for impulsive or impact problems, where velocities can undergo a sudden jump. They also proposed the Stochastic Integrity Measure (SIM), defined as “the mean escape time when the attractor

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is subjected to white noise of prescribed intensity”, which is of course relevant for problems with stochastic excitation. Stochastic basins of attraction have been considered, e.g., in [87]. When the attractor is a periodic solution of a continuous time system, i.e. a point in the associated Poincaré map, LIM and IIM are relatively easy to determine. But for quasi-periodic and, mostly, chaotic attractors, its determination is cumbersome and very onerous from a numerical point of view. Furthermore, it is strongly related to the presence of an attractor, and thus does not apply when the safe basin is not a basin of attraction (e.g., when transient dynamics are relevant). To overcome the previous drawbacks, the Integrity Factor (IF) has been introduced [59, 117]. It is the radius of the largest hyper-sphere entirely belonging to the safe basin, irrespective of the position of its center. It is clearly a measure of the basin only, not requiring any attractor, it is easy to determine and, as LIM and IIM, rules out the fractality in the evaluation of robustness. Both LIM and IF can be determined only numerically, the latter being more difficult to compute, starting from the previously determined basin/attractor and safe basin, respectively. They do not have a theoretical background allowing for a deeper understanding, although some properties are reported in Sect. 11.4.4. Clearly it is LIM.≤IF. Various improvements have been successively proposed. Safe basins are always obtained in a discrete way, i.e. by approximating the continuous phase space by a finite number of cells (pixels). The actual safe basin is the set obtained by eliminating the cells which are not surrounded by cells of initial conditions leading to another attractor [57, 90]. In other words, the actual safe basin is obtained by removing the boundary points from a given safe basin. This clearly eliminates from the basin the fractal parts (where points border with other basins), and thus the actual safe basin is made mainly by the compact part of the basin. Thus, we no longer need to pay attention to the definition of the measure to rule out fractality, and we can take the easiest one, namely the magnitude of the actual safe basin, thus obtaining the so-called Actual Global Integrity Measure (AGIM). This evidently satisfies the inequality AGIM.≤GIM. The next development is related to the fact that, for continuous time systems, the safe basin is obtained for a given Poincaré section, and its properties may vary by varying the section along the orbit. This is particularly relevant for periodic systems, in which the stroboscopic Poincaré section depends on the phase of the excitation. Since it is commonly unknown, it is convenient to have phase-independent basins, that are obtained as the intersection of all safe basins by varying the phase of the excitation. This leads to the so-called true safe basin [59, 115], that is the set over which we have to apply an integrity measure. In [116] an algorithm is proposed to simplify the computation of true safe basins of attraction, that otherwise is very demanding. Since commonly the true safe basin is compact, like the actual safe basin, the GIM applied to this set, i.e. the True Global Integrity Measure (TGIM), is a reliable measure of robustness. Finally, it is noted that in the phase space not all variables have the same relevance and the same physical dimension (e.g. displacements vs velocities). So, considering,

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as in the definition of stability, in LIM and in IF, only a hyper-sphere in the phase space could be limitative since this gives the same weight to all variables. To overcome this drawback, the Anisometric Local Integrity Measuse (ALIM) and the Anisometric Integrity Factor (AIF) have been defined in [62]. They are just the extensions of LIM and IF, respectively, but considering a distance that weights in different manner each coordinate, and in particular coordinates with different physical dimensions. In the simple 2D case the anisometric distance is d=

.

√

x 2 + β x˙ 2 ,

(11.31)

where .β is the anisometric parameter and .x = x1 − x2 is the difference between two points in the phase space. The extension to larger dimensional systems is immediate. .β = 0 corresponds to .d = |x|, .β = 1 to LIM and IF, and .β → ∞ to .d = | x|, ˙ i.e. to IIM. There are two different approaches. In the first, .β is fixed a priori, i.e. the different relevance of each variable known. In this case the determination of ALIM and AIF is identical to that of LIM and IF. On the contrary, there are cases in which this information is not known, and one needs to get it from the simulations. Here, one can use .β as an extra variable, and look for the largest ellipsoid (by varying .β) satisfying the LIM or IF definition. Now the outcome is not only the integrity measure of the basin, but also the different relevance of each variable. A possible extension of (11.31) consists of considering the distance / d=

.

n

˙ n , n ≥ 1. |x|n + β |x|

(11.32)

For .n = 1 it is a rhombus, for .n = 2 a circle and for .n → ∞ a square, i.e. .d tends to the weighted sup-distance.

11.4.3 Integrity Profiles All the developments summarized so far refer to fixed values of the parameter .µ of (11.1). It is important to see what happens when .µ is varying, i.e. to perform a parametric analysis. When referring to classical Lyapunov stability, this leads to the local bifurcation theory [19–21] that, roughly speaking, is aimed at studying when attractors lose local stability. In parallel, also global bifurcations have been investigated, and concepts like crises [118, 119], catastrophes [120–122], etc. have been introduced. Here there is a sudden modification of the attractor not due to local loss of stability, but due to global events commonly triggered by appropriate homoclinic or heteroclinic bifurcations. In this section I discuss what happens to the integrity measures IM (any one of those defined in Sect. 11.4.2) when one parameter .μ of the system varies. The function IM(.μ) is named integrity profile and provides a description of what happens

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325

to the robustness by varying .μ. In particular, it immediately shows for which values of .μ the IM is high, i.e. the system is safe, and for what values of .μ the IM is low, i.e. the system, while stable, is unsafe and dangerous. It is just in these regions that dynamical integrity is an indispensable tool to be added to the classical stability. It is better to illustrate the main properties of integrity profiles referring to a specific example. Here I consider the Duffing equation .

x¨ + 0.1x˙ − x + x 3 = A sin(1.2t)

(11.33)

(see Fig. 11.12), and consider the excitation amplitude . A as the varying parameter. The safe basin is the classical basin of attraction in the stroboscopic Poincaré section at .t = 0. The conclusions drawn with this specific example are general, and have been observed in many other, even more complex, systems. Because of the spatial symmetry of the considered system, we have the same behaviour on the right and on the left potential well. Thus, I illustrate only the results for the right part of the phase space,.x ≥ 0. Here there are two main period 1 attractors, resonant (R) and nonresonant (NR) ones (see Fig. 11.13). Limiting to GIM, LIM and IF, the two associated integrity profiles are reported in Fig. 11.14. For low values of . A all integrity measures maintain their initial value. There is no relevant reduction of integrity for increasing excitation amplitude. When R is born (by a saddle-node local bifurcation at . A ≃ 0.05373, see Fig. 11.13), the integrity of the NR decreases, while the integrity of R starts from zero. Details of what happens in this case are discussed in Sect. 11.4.4.

1.8

x

SN

PD

SN

0 0

A

Fig. 11.13 Bifurcation diagram of (11.33). The upper (from . A = 0 to the upper SN) black path is the nonresonant (NR) attractor, while the lower (from the lower SN to PD) is the resonant (R) one

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GIM

326

NR R 0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

A

NR

LIM

R

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

A NR

IF

R

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

A Fig. 11.14 Integrity profiles of GIM, LIM and IF of the resonant (R) and non resonant (NR) attractors of (11.33)

In the range of coexistence between R and NR there is an exchange of robustness: NR loses it, while R increases it. However, while with GIM and IF the R attractor recuperates “all” the integrity lost by NR, with LIM it recuperates only a part, and after the disappearance of NR it has about 40% of the initial integrity. The marked quantitative difference between IF and LIM is due to the fact that the attractors are not around the center of their basins. In this case LIM is the most conservative, and thus most reliable, measure of integrity. After the disappearance of NR (by a reverse saddle-node local bifurcation at . A ≃ 0.1017, see Fig. 11.13) the major difference between GIM, from one side, and

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LIM and IF, from the other side, appears. In fact, here GIM is almost constant, while both LIM and IF decrease. This is due to the fractality of the basins, just as illustrated in Fig. 11.12, and confirms that in this range GIM is not reliable, while both LIM and IF are trustable, although to a different extent. In this interval of . As the LIM and IF have low values, and go to zero when R disappears, at . A ≃ 0.1987, at the end of a period doubling cascade started with the first period doubling bifucation at . A ≃ 0.180 and ended by a boundary crisis (see Fig. 11.18). In particular I highlight that for . A > 0.140, both LIM and IF have values less than 20% of the initial value at . A = 0. This means that here the basins are very fractal (see Fig. 11.12b), or eroded, and the R attractor, while remaining stable in the Lyapunov sense, is not robust, unsafe and practically unstable. It is just in this region that without dynamical integrity one loses fundamental information on safeness. The safe critical threshold is . A ≃ 0.14, much less than the theoretical one . A ≃ 0.1987 (if one refers to the disappearance of the in-well attractor), or . A ≃ 0.180 (if one refers to the loss of local stability of the period 1 attractor). Another property illustrated in Fig. 11.14 is that close to local saddle-node bifurcations, where attractors appear/disappear, the integrity measures are small, i.e. the safe basin is small. This means that it will be almost impossible in practical applications to reach the exact value of . A where the bifurcation occurs, which however remains as the theoretical driving phenomenon. The second general conclusion is that dynamical integrity is needed around saddle-node bifurcations to understand possible discrepancies between theoretical and experimental results [90, 93]. Finally, the last general conclusion is that integrity profiles immediately highlight the region of parameters where the dynamical integrity is necessary and classical stability is not enough. The strong oscillation of GIM observed for large values of . A is mainly given by the birth and sudden disappearance of minor attractors. Examples are reported in the next section (see in particular Fig. 11.17).

11.4.4 On the Regularity of Integrity Profiles When a new attractor N is born (by a local saddle-node bifurcation) inside the basin of a previously existing P attractor (for example at . A ≃ 0.05373 in Fig. 11.13, where N is the resonant R and P is the non resonant NR, but also at . A ≃ 0.1017), the integrity of the P decreases. GIM decreases suddenly but continuously, while LIM and IF have a jump. This is due to the fact that the new basin of N grows from zero (as confirmed by the GIM of N) so that GIM is continuous, but it is born inside the basin of P, so that the largest circles involved in the definitions of LIM and IF instantaneously reduce (see Fig. 11.15). This is the first general conclusion. The birth of a new solution (by a saddle-node bifurcation) can generate a continuous fall of GIM and, in general, a sudden jump of

328

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1

0

0

-1 -1.8

0

1.8

-1 -1.8

(a)

0

1.8

(b)

Fig. 11.15 Basins of attraction of the stroboscopic Poincaré map of (11.33). Black circle is IF, blue circle is LIM, the dot is the period 1 solution. a . A = 0.053; b . A = 0.054 1

1

0

0

-1 -1.8

0

(a)

1.8

-1 -1.8

0

1.8

(b)

Fig. 11.16 Basins of attraction of the stroboscopic Poincaré map of (11.33). Black circle is IF, blue circle is LIM, the dot is the period 1 solution. a . A = 0.144; b . A = 0.145 with the minor attractor (in grey)

LIM and IF—and thus an instantaneous loss of robustness—of previously existing solutions. This in particular happens for minor attractors that, although likely being not so important by themselves (since they exist only for a small range of parameters and have a small basin of attraction), strongly reduce the robustness of the main attractor. An example is shown in Fig. 11.16. However, it may happen that the new attractor N (minor or not) is born not in the compact part of P, and in this case it affects GIM but not LIM and IF (see Fig. 11.17).

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1

1

0

0

-1 -1.8

0

1.8

-1 -1.8

0

1.8

(b)

(a)

Fig. 11.17 Basins of attraction of the stroboscopic Poincaré map of (11.33). Black circle is IF, blue circle is LIM, the dot is the period 1 solution. a . A = 0.156; b . A = 0.158 with the new attractor (in grey) out of the compact part Fig. 11.18 Boundary crisis at . A = 0.1987 on the stroboscopic Poincaré map of (11.33)

-0.5

-0.525 0.64

0.83

As a final remark on the continuity of the curves, I recall that in correspondence of a crisis the (chaotic) attractor suddenly disappears. Thus, trivially, any integrity measure jumps down to zero, and thus has a discontinuity. An example of boundary crisis is reported in Fig. 11.18, where it is clear that the attractor (in black) touches the boundary of its basin (in green).

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At the saddle-node, while continuous, the curve GIM(. A) has a singular behaviour, in particular the derivative goes to infinity. This is confirmed by the least square approximation of the GIM(. A) of NR, that provides the estimates GIM(A) = GIM(Acr ) − F1 (A − Acr )0.19 + ...,

.

(11.34)

around . Acr ≃ 0.05373, and GIM(A) = F2 (Acr − A)0.27 + ...,

.

(11.35)

around. Acr ≃ 0.1017, where. F1 and. F2 are two positive numbers that are not relevant here. The fact that the exponent is not the same for the two saddle-nodes implies that there is the same qualitative behaviour, but not the same quantitative behaviour. In other words, there is no a quantitative universal law for GIM approaching a saddlenode. I now focus on the derivatives of LIM(. A), and I refer to the case of period 1 attractor for simplicity. An example is clearly visible at . A ≃ 0.65 in Fig. 11.15b for the R attractor. The LIM is given by |axc + b − x˙c | , .LIM(A) = (11.36) √ a2 + 1 where .(xc , x˙c ) are the coordinates of the attractor on the Poincaré section, while .a and .b are the coefficients of the line .x˙ = ax + b that in the phase space is tangent to the boundary of the basin in the point where the circle of LIM touches the boundary. All depend on . A, and thus ] [ ' dLIM a xc + axc' + b' − x˙c' ' ax c + b − x˙ c ' = LIM = sign(axc + b − yc ) . −aa 2 . √ dA (a + 1)3/2 a2 + 1 (11.37) When . A is varying, since the system is smooth, .xc and .x˙c vary smoothly, so that their derivatives are continuous. However, when the circle-basin boundary touching point changes, we have that .a and .b change, and thus their derivatives .a ' and .b' , so that we have a discontinuity of the derivative of LIM(. A). This phenomenon is illustrated in Fig. 11.19. The IF is given by √ IF(A) =

.

(x1 − x2 )2 + (x˙1 − x˙2 )2 , 2

(11.38)

if the largest circle touches the boundary in only two points .(x1 , x˙1 ) and .(x2 , x˙2 ) (see Fig. 11.20a), and by

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1

1

0

0

-1 -1.8

0

1.8

(a)

-1 -1.8

0

1.8

(b)

1

0

-1 -1.8

0

1.8

(c) Fig. 11.19 Basins of attraction of the stroboscopic Poincaré map of (11.33) across the point . A = 0.065 of discontinuity of LIM(. A). Blue circle is LIM, the dot is the period 1 solution. a . A = 0.062; b . A = 0.065; c . A = 0.070

√ .IF(A)

=

√ √ (x1 − x2 )2 + (x˙1 − x˙2 )2 (x1 − x3 )2 + (x˙1 − x˙3 )2 (x2 − x3 )2 + (x˙2 − x˙3 )2 , 2|x˙1 (x3 − x2 ) + x˙2 (x1 − x3 ) + x˙3 (x2 − x1 )|

(11.39) if the largest circle touches the boundary in three points .(x1 , x˙1 ), .(x2 , x˙2 ) and .(x3 , x˙3 ) (see Fig. 11.21a). The coordinates of these points smoothly depend on . A. The continuity of. dIF = IF' can be studied in a similar way of that of LIM, although dA the formulas are much more involved. Now there is a discontinuity of the derivative when (i) the two touching points change, (ii) we pass from two to three circle-basin boundary touching points, as reported in the example of Fig. 11.20, or (iii) when the three touching points change, see the case of Fig. 11.21.

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0

1.8

-1 -1.8

0

1.8

(b)

(a) 1

0

-1 -1.8

0

1.8

(c) Fig. 11.20 Basins of attraction of the stroboscopic Poincaré map of (11.33) across the point . A = 0.080 of discontinuity of IF(. A). Black circle is IF. a . A = 0.075; b . A = 0.080; c . A = 0.085

11.5 Conclusions and Further Developments The long history that goes from equilibrium to stability to dynamical integrity has been summarized, according to the point of view, and in some sense the taste, of the author. After recalling what continuous and discrete time dynamical systems are, the different kinds of motion—equilibrium/fixed point, periodic, quasi-periodic, homo/ heteroclinic and chaos—have been recalled. Then, the main results of the classical stability theory have been reported. The principal drawbacks of stability have been highlighted, and the necessity to introduce the dynamical integrity is shown. As a matter of fact, this requires passing from a local

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1

1

0

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0

1.8

-1 -1.8

0

1.8

(b)

(a) 1

0

-1 -1.8

0

1.8

(c) Fig. 11.21 Basins of attraction of the stroboscopic Poincaré map of (11.33) across the point . A = 0.1108 of discontinuity of IF(. A). Black circle is IF. a . A = 0.1106; b . A = 0.1108; c . A = 0.1120

to a global analysis of the dynamical system, and thus it is much more complicated, even if it provides the necessary information for robustness and safeness. The definition of safe basin and different integrity measures have been recapitulated. Then, attention was paid to the integrity profiles, that describe how the robustness varies by changing a parameter of the system. They are the extension to dynamical integrity of the well-known bifurcation diagrams. Some new results related to the continuity and smoothness of erosion profiles have been reported, too. The aim of this chapter is not to present completely new material, even if there are some new results (see in particular Sect. 11.4.4), but rather to provide the reader a “complete” summary of concepts and tools needed to evaluate the robustness of given attractors, which is the necessary pre-requisite for their safe and cognizant use in practical applications.

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I believe that the main ideas needed for a dynamical integrity analysis have been more or less introduced in previous works (and here summarized), even if of course various further theoretical developments are possible and worthwhile, indeed. The big challenge for the future is to apply these concepts to high dimensional systems, where one faces the formidable problem of computing basins of attraction. In fact, to compute the basins of 6D systems is at this time extremely difficult, while going to 8D and over looks almost impossible both in terms of computational and memory efforts for available computers. Attempts are in progress [52–54]. Accepting the determination of the integrity measure directly, without resorting to the full knowledge of basins (which is of course not welcome because it means a lack of understanding), will help to simplify the problem a bit, but not to completely overcome it. Acknowledgements This paper summarizes the work I have done on dynamical integrity in the last decades together with Giuseppe Rega and other collaborators, among which my former PhD students Pierpaolo Belardinelli, Laura Ruzziconi and Nemanja Andonovski. I am extremely in debt to all of them.

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