Energy transfer and dissipation in nonlinear oscillators Maaita Tzamal- Odysseas Physicist, MSc. Computational Physics A thesis submitted for the degree of Doctor of Philosophy in the Department of Physics School of Sciences Aristotle University of Thessaloniki Supervisor: Assistant Professor Efthymia Meletlidou March, 2014 To my grandmothers Mdallal Enayat and Polidou Marika 3 Abstract (In Greek) Sthn παρούσα didaktορική διατριβή μελετάμε èna σύσthma tri¸n suzeugmènwn talantwt¸n me τριβή, dύο grammik¸n me ènan mh γραμμικό. Tètoia susτήμαta talantwt¸n èqoun μεγάλο endiafèron idiaÐtera όtan h μάζα tou mh γραμμικού talantωτή eÐnai poλύ μικρόterh από touc γραμμικούc me sunèpeia o mh γραμμικός talantωτής na leitourgeÐ wc katαβόθρα enèrgeiac. Autού tou eÐdouc ta susτήματa, sta opoÐa συνυπάρχουν ènac αργός kai ènac γρήγορος χρόnoc mpoρούν na μελετηθούν me th βοήθεια thc singularity analysis, twn analloÐwtwn pollaploτήτwn, en¸ σημαντική plhroforÐa gia th δυναμική tou susτήματoc dÐnetai kai από thn δυναμική thc αργής ροής (Slow Flow) tou susτήματoc. Sthn παρούσα διατριβή melετάμε to σύσthma mèsw thc melèthc thc αργής analloÐwthc poλλαπλόthtac (Slow Invariant Manifold- SIM-). Me th βοήθεια tou θεωρήματoc tou Tikhonov kathgoriopoiούμε tic διάφορες peript¸seic thc αργής analloÐwthc poλλαπλόthtac kai orÐzoume αναλυτικά tic συνθήκες me tic opoÐec μπορούμε na οδηγηθούμε sthn κάθε perÐptwsh. Se επόμενο βήμα μελετάμε thn δυναμική thc argήc rοής kai παρατηρούμε όti h δυναμική thc eÐnai πλούσια αφού oi troqièc thc μπορούν na eÐnai kanonikèc, na κάνουν talant¸seic hremÐac (relaxation oscillations), ή na eÐnai qaotikèc. Από thn melèth thc enèrgeiac pou αποθηκεύετai ston mh γραμμικό talantωτή kai από ton ρυθμό απόσβεσης thc sunoλικής enèrgeiac tou susτήματoc παρατηρούμε όti tόσο h ύπαρξη diaklad¸sewn thc argής analloÐwthc poλλαπλόthtac, όσο kai h dunaμική thc αργής ροής paÐzoun kajorisτικό ρόlo sthn metαφορά enèrgeiac apό ton γραμμικό ston mh γραμμικό talantωτή. EpÐshc, stic peript¸seic pou blèpoume metαφορά enèrgeiac παρατηρούμε όti o ρυθμός apόσβεσης thc sunoλικής enèrgeiac tou susτήματoc eÐnai megaλύτερος από ton ρυθμό απόσβεσης όtan den metafèretai enèrgeia ston mh γραμμικό talantωτή. H melèth tou susτήματoc twn tri¸n suzeugmènwn talantwt¸n kleÐnei me thn πρόtash ενός mh γραμμικού hlektrikou kukl¸matoc to opoÐo ulopoieÐ thn mh γραμμική διαφορική exÐswsh δεύτερης τάξης me thn opoÐa proseggÐsame to αρχικό sύsthma. To sugkekrimèno κύκλωμα èqei endiafèron giatÐ mac dÐnei th δυνατόthta na μελετήσουμε kai πειραματικά διάφορα apό ta faiνόμενα pou eÐdame sthn θεωρητική mac ανάλυση. 5 Prologue I was introduced to the subject of this thesis by the late Associate Proffessor of the department of Physics, Simos Ichtiaroglou. Simos Ichtiaroglou participated in the second International Conference on Nonlinear Normal Modes and Localization in Vibrating Systems that took place in Samos at 2006, where he attended a number of paper presentations related to targeted energy transfer. So, when we were discussing the topic of my thesis he proposed to me and to Assistant Professor Efi Meletlidou to work on energy transfer and dissipation because he considered it interesting. The topic was indeed interesting. It included rich theoretical study and practical applications, so we decided to start work on it. The specific system composed of two coupled linear oscillators and a nonlinear oscillator that interacts through an essential nonlinearity with one of the linear oscillators was proposed by proffessor Alex Vakakis in the Euromech Colloquium 503 - Nonlinear Normal Modes, Dimension Reduction and Localization in Vibrating Systems- that took place in Frascati(Rome) at 2009. The work on this thesis was both interesting and challenging. It was interesting because I came into contact with new theories and tools, such as the singular perturbation theory, the multiple scale analysis, the complexification- averaging technique etc... that I was not taught neither in my undergraduate nor graduate studies. It was also interesting because I came into contact with other scientific fields, such as engineering, and I learned a lot of new things, as for example, how we can implement a nonlinear-cubic- term in a mechanical structure. It was challenging, mainly, because there was no financial support and I was obliged to work all the years of my doctoral dissertation outside the university. Our work resulted in four publications in peer reviewed international journals. The first paper is titled “The effect of slow flow dynamics on the oscillations of a singular damped system with an essentially nonlinear attachment” and it was published in the Journal of Applied Nonlinear dynamics in 2013. The second paper is titled “Analytical Homoclinic Solution of a Two-Dimensional Nonlinear System of Differential Equations” and it was published in the 7 Journal of Nonlinear Dynamics in 2013. The third paper is titled “The Study of a Nonlinear Duffing - Type Oscillator Driven by Two Voltage Sources” and it was published in Journal of Engineering Science and Technology Review in 2013. Finaly the fourth paper is titled “The dynamics of the slow flow of a singular damped nonlinear system and its Parametric Study” and it is accepted for publication in the Journal of Applied Nonlinear Dynamics. This doctorate thesis marks the end of a stage and the begining of a new one in my life. It is therefore time for reasoning. I want to thank my supervisor, Assistant Professor, Efi Meletlidou, for helping me to become a researcher, for teaching me many things, especially on mathematics. She was always understanding and patient and despite the difficulties, we managed to cooperate well. I want to thank my advisor, Associate Professor Iannis Kyprianidis, for his support all these years. His help was important for the implementation of the electric circuit, for the numerical work on computing the maximal Lyapunov Characteristic Exponent. I want to thank my advisor, Professor Alex Vakakis, for the proposal of the system and his guidance on its study. His encouraging words always gave me strength and courage to continue. I want to thank, Dr. Christos Volos, for his help on the inplementation and study of the electric circuit and teaching me the Multisim program. I want to thank, Associate Professor Vassilis Rothos, for his advice on books and papers. Professor George Tsaklidis was a mentor on how to write, check and correct a scientific manuscript. He is a good advisor and a friend. I want to thank, Assistant Professor Menios Tsiganis, for his help and advice, especially, for the Lyapunov Characteristic Exponents and the numerical work. We always have good debates and discussions. I want to thank, Professor Harry Varvoglis, for being there from the beginning of my PhD. Although we disagreed on different issues, he was always there, honest and taking an interest. Assistan Professors George Voyatzis and Harry Skokos, and Special Teaching Fellow Fotini Zervaki, thank you for the support on difficult times. I want to thank, Professor Antonis Anagnostopoulos, for his advice and remarks. Special thanks to my “officemates”, Dr. Stella Tzirti and PhD Candidate Kallinikos Nikos, for the good collaboration and the constructive discussion all these years. I want to thank my friend, PhD Candidate Nikos Bastas, for his help on Grid computing. I want to thank, from the depths of my heart, my parents Ifigeneia and Sami Maaita for all the things that they have done to let me become what i am today. My mother for being by my side, working with me on school, supporting me on difficult times. My father for his guidance, rational thinking and encouragment. Also, my brother Tarek Maaita for always being there when I need him. My wife Kiki Moysiadou for her constant support, for her critical remarks and the continuous 8 interaction. For giving birth to my two beloved little girls Tereza- Leila, Ifigeneia- Yasmin and the one that we are waiting to come next June. Finaly, I want to thank all those who contributed in one way or another in the formation of my being. Maaita Jamal- Odysseas, Thessaloniki, Sunday 30th March, 2014. 9 Contents 1 Introduction 15 2 Basic concepts, theorems and tools 21 2.1 Dynamical Systems................................ 21 2.1.1 General definitions ............................ 21 2.1.2 Equilibrium points (Periodic orbits) and stability: definitions and theorems 22 2.1.3 Bifurcation theory............................. 25 2.1.4 Chaos................................... 27 2.1.5 The Averaging method.......................... 29 2.2 Singular Perturbation Theory........................... 30 2.2.1 What is a singular perturbation problem? ................ 30 2.2.2 Multiple Scale Analysis.......................... 30 2.2.3 Geometric Singular Perturbation Theory................. 33 2.3 Electric circuits.................................. 36 2.3.1 Linear and nonlinear electric elements.................. 36 2.3.2 Resistor connections and basic laws ................... 39 2.3.3 Operational Amplifier........................... 40 2.3.4 Examples of nonlinear circuits...................... 41 3 Classification of the SIM 47 3.1 Reduction of the system.............................. 48 3.2 Singular Perturbation Analysis.......................... 51 3.3 Solutions and numerical results.......................... 54 3.4 Conclusions.................................... 57 4 The dynamics of the slow flow 61 4.1 The dynamics of the slow flow.......................... 61 4.2 Conclusions.................................... 71 10 5 Energy transfer and dissipation