Huygens Synchronization in Various Dynamical Systems - Experimental Results D.J. Rijlaarsdam

DCT 2008.049 June 2008

Graduation Committee Prof. dr. H. Nijmeijer (supervisor) Dr. A.Yu. Pogromsky (coach) Dr. ir. N. Van de Wouw Master’s Thesis Dynamics & Control Group Eindhoven University of Technology Eindhoven, June 2008

Things should be made as simple as possible, but not any simpler. [Albert Einstein]

Abstract

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Abstract Synchronization phenomena take place frequently and all around us., Synchronization of sleep rhythms to the rhythm of day and night, for example, is familiar to everyone. Less known, but no less important, is the synchronization in neuronal activity that allows the brain to function as it does. Apart from biological synchronization, the electrical synchronization taking place when turning on a radio or using a mobile phone, is also a frequent phenomenon in our everyday life. Although it is clear that synchronization is a rather common phenomenon, the mechanisms behind synchronizing phenomena are far from straightforward. A better understanding of these mechanisms is required to understand - and possibly control - the related processes such as epileptic seizures or high-end encrypted communication. Many theoretical and experimental studies have been performed, leading to a better understanding of these synchronization phenomena. One of the first to notice and study such phenomena in detail was Christiaan Huygens, whos observations are strongly related to the research presented in this thesis. This thesis aims to add to this field of knowledge by supplying additional and new analytical, numerical and experimental results. These results are generated by using a new experimental set-up developed at the Eindhoven University of Technology. Using this set-up, both controlled and uncontrolled synchronization between a variety of different oscillators can be investigated. The main research objective of this thesis is to: Investigate, using a new experimental set-up, the existence and stability of synchronization regimes in coupled oscillatory systems. This objective is realized by initially focusing on the experimental set-up. This set-up, consists of two oscillators connected to a common beam by leaf springs. The beam itself is supported by leaf springs as well, thus allowing coupling between the oscillators by displacement of the beam. After a detailed discussion of the set-up, a variety of synchronization phenomena is presented, showing synchronization taking place in a range of dynamical systems, such as coupled Duffing oscillators and coupled rotating discs. The first part of the thesis describes the experimental set-up and the derivation and identification of a model for this set-up in detail. The possible synchronization regimes in the dynamics of the set-up are investigated as well. Furthermore, a stability analysis and numerical as well as experimental results are presented. These results show the attractive and stable nature of the anti-phase synchronization manifold in the dynamics of the set-up. The experimental set-up described in part 1 of the thesis is fully actuated. This property is used in the second part of the thesis by designing a state feedback controller which allows the modeling of a variety of dynamical systems. To illustrate this, experiments are conducted while modulating the set-up so as to respectively resemble a system of coupled Duffing oscillators or rotating discs. For each dynamical system, experimental results are preceded by a stability analysis of the synchronization manifold and simulation results. Finally, a controller similar to the one used for the Duffing oscillators and the rotating discs, is developed in order to imitate the classical Hygens set-up, which consists of two pendula mounted on a common frame. With the Huygens set-up no successful experimental results are obtained, however. The most probable cause for this is that, due to the limited actuator power in the system, the modeled coupling is weak compared to the disturbances. Therefore, no synchronization has been observed yet. This thesis, therefore, describes, in detail, the theoretical analysis of Huygens’ system and gives an extensive analysis of the observed problems as well as their possible solutions.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

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Samenvatting

Samenvatting Gedurende het dagelijks leven heeft iedereen regelmatig te maken met synchronisatie. Het feit dat u bijvoorbeeld automatisch ’s ochtends wakker wordt, wordt veroorzaakt doordat het slaapritme zich aanpast aan, oftewel synchroniseert met, het ritme het dag en nacht. Daarnaast is synchronisatie tussen neuronen in de hersenen verantwoordelijk voor de verwerking van de tekst die u hier leest. Naast synchronisatie in biologische systemen vindt synchronisatie plaats bij het aanzetten van een mobiele telefoon of een radio, waaruit tevens blijkt hoe gewoon dergelijke effecten in het alledaagse leven zijn. Hoewel synchronisatie een breed geaccepteerd en veel benut fenomeen is zijn de onderliggende mechanismen lang niet zo eenvoudig als soms lijkt. Een beter begrip van de achterliggende mechanismen kan echter leiden tot een beter begrip van de effecten zelf, zoals bijvoorbeeld epileptisch aanvallen of geavanceerde encryptie methoden. In het verleden zijn er reeds vele theoretische en experimentele studies aan dit onderwerp gewijd. Een van de eerste en belangrijkste observaties van synchronisatie werd in 1665 opgetekend door Chistiaan Huygens. Zijn resultaten vormen een belangrijk deel van de basis van het onderzoek dat in dit rapport gepresenteerd wordt. In dit afstudeerproject worden nieuwe analytische en experimentele resultaten gepresenteerd met betrekking tot (Huygens) synchronisatie. Voor dit onderzoek is gebruik gemaakt van een speciaal ontworpen opstelling bij de faculteit Werktuigbouwkunde van de Technische Universiteit Eindhoven. De hoofdvraag die gedurende dit afstudeertraject bestudeerd is luidt: Onderzoek, met gebruik van een nieuwe experimentele opstelling, het bestaan en de stabiliteit van synchronisatie regimes in gekoppelde oscillerende systemen. Om te beginnen wordt aandacht besteed aan de, speciaal voor dit onderzoek ontworpen, opstelling. Deze opstelling bestaat uit twee oscillatoren die beide met behulp van bladveren aan een gemeenschappelijke balk zijn gekoppeld. Deze is op zijn beurt tevens vrij opgehangen in een set bladveren. Zodoende wordt de beweging van de oscillatoren gekoppeld via een beweging van de balk. Na een bespreking van het ontwerp en de eigenschappen van deze opstelling wordt nader ingegaan op de synchronisatie tussen de oscillatoren, zoals die tijdens experimenten naar voren komt. Voor deze experimenten wordt de opstelling onder andere door actuatie aangepast om naast met de natuurlijke dynamica van het systeem, ook experimenten te kunnen uitvoeren met andere dynamische systemen. In het speciaal wordt ingegaan op de synchronisatie tussen twee gekoppelde Duffing oscillatoren en een systeem van twee gekoppelde roterende schijven. Het eerste deel van het afstudeerverslag beschrijft de afleiding en identificatie van een model voor de opstelling. Een analyse van mogelijke synchronisatieregimes in de dynamica van de opstelling wordt tevens gepresenteerd. Naast experimentele resultaten worden simulaties en analytische resultaten gepresenteerd die de stabiliteit van de geobserveerde synchronisatieregimes onderstrepen. In de experimentele opstelling zoals die in het eerste deel van het afstudeerverslag ge¨ıntroduceerd wordt kan op alle graden van vrijheid geactueerd worden. Deze eigenschappen worden benut door een state feedback controller te ontwerpen waarmee de dynamica in de opstelling aangepast kan worden, zodat deze een ’willekeurige’ set dynamische vergelijkingen representeert. Er wordt in detail ingegaan op drie casestudies: Ten eerste wordt synchronisatie bestudeerd in een systeem van gekoppelde Duffing oscillatoren. Vervolgens wordt een gelijksoortige analyse gepresenteerd van een systeem van gekoppelde roterende schijven. In beide gevallen worden de experimentele resultaten vooraf gegaan door een stabiliteitsanalyse van de mogelijke synchronisatieregimes en simulatie resultaten. De derde casestudie behelst die van het originele systeem zoals dat destijds door Christiaan Huygens werd gebruikt. Tot op heden heeft deze studie helaas nog geen succesvolle experimentele resultaten opgeleverd. De meest waarschijnlijke reden hiervoor is dat de koppeling in het geval van deze casestudie erg klein is ten opzichte van verstoringen. Voor deze casestudie wordt in plaats van experimentele resultaten een gedetailleerde analyse gegeven van de geobserveerde problemen en hun mogelijke oplossingen.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

Contents

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Contents 1 Introduction

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Huygens 2008

2 History and Motivation 15 2.1 A Brief History of Christiaan Huygens . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 300 Years of Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3 Set-up 3.1 Set-up Design . . . . . . . . . . . . . . . . . . . . . . . 3.2 Technical Discussion of the Set-Up . . . . . . . . . . . 3.2.1 Data Acquisition, Safety System and Software 3.2.2 Tuning the Actuators and Sensor Calibration . 3.2.3 Viscous Damping and Hysteresis . . . . . . . .

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4 Modeling the Set-Up 4.1 Equations of Motion . . . . . . . . . . . . . . . . . 4.2 System Identification . . . . . . . . . . . . . . . . . 4.2.1 Design of the Excitation Signal . . . . . . . 4.2.2 Identification of the Individual Subsystems 4.2.3 Coupling Parameters . . . . . . . . . . . . . 4.2.4 Nonlinear Stiffness . . . . . . . . . . . . . . 4.3 Concluding Remarks . . . . . . . . . . . . . . . . .

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5 Analysis and Simulation Results 5.1 Synchronization: Introduction and Definition 5.2 Stability of the Synchronization Manifold . . 5.3 Simulation Results . . . . . . . . . . . . . . . 5.4 Experimental Validation . . . . . . . . . . . . 5.5 Concluding Remarks . . . . . . . . . . . . . .

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Experimental Results

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6 Masking of System Dynamics 7 Coupled Duffing Oscillators 7.1 Equations of Motion and Stability Analysis 7.2 Simulation Results . . . . . . . . . . . . . . 7.3 Experimental Results and Controller Design 7.4 Concluding Remarks . . . . . . . . . . . . .

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8 Coupled Rotating Discs 8.1 Equations of Motion and Controller Design . . . . . 8.2 Rotating Discs without Eccentric Masses . . . . . . . 8.2.1 Simulation and Experimental Results . . . . 8.2.2 Convergence Rate (Analysis and Simulation) 8.3 Rotating Discs with Eccentric Masses . . . . . . . . 8.3.1 Simulation and Experimental Results . . . . 8.4 Concluding Remarks . . . . . . . . . . . . . . . . . .

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Contents

9 Huygens Clocks 52 9.1 Equations of Motion and Controller Design . . . . . . . . . . . . . . . . . . . . . . . 52 9.2 Problems and Solutions Concerning Experimental Results . . . . . . . . . . . . . . . 54 9.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 10 Conclusions and Recommendations 57 10.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 10.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 A Proofs and Derivations 63 A.1 Equations of Motion (Lagrange) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 A.2 Set-up: Stability of the Synchronization Manifold . . . . . . . . . . . . . . . . . . . . 65 A.3 Rotating Discs: Stability of the Synchronization Manifold . . . . . . . . . . . . . . . 67 B Additional Data: System Identification B.1 Parameter Values . . . . . . . . . . . . . . . . . . . . . B.2 Phase Shifts Obtained by the Infinity Norm Algorithm B.3 Nonlinear Force-Displacement Characteristics . . . . . B.4 Additional Figures . . . . . . . . . . . . . . . . . . . .

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C Technical Specifications C.1 Connection Scheme Set-Up . . . . . . . C.2 Safety System . . . . . . . . . . . . . . . C.3 Masses and Eigenfrequencies of Set-up . C.4 Data Acquisition Systems . . . . . . . . C.5 Relative Calibration of the Voice Coil . C.6 Damping Analysis . . . . . . . . . . . . C.6.1 Viscous Damping . . . . . . . . . C.6.2 Hysteresis and Dry Friction . . . C.7 Linear Variable Differential Transformer C.8 Linear Motor (Voice Coil Actuators) . .

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D Related Papers and Conferences

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Huygens Synchronization in Various Dynamical Systems. Experimental Results.

Preface

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Preface This thesis presents the results of my graduation project at the Dynamics and Control group at the department of Mechanical Engineering (Eindhoven University of Technology)1. The project deals with the theoretical and experimental investigation of synchronization phenomena in a range of dynamical systems using a novel set-up developed at the departement. The first part of the project is devoted to the development of the necessary hard- and software and the identification the dynamics present in this set-up. The second part focusses on investigating synchronizing behaviour in a range of dynamical systems by means of analysis, simulations and experiments. I would like to take this opportunity to thank the people who made this project possible. First of all, I would like to thank the members of my graduation committee: prof. dr. H. Nijmeijer, dr. A. Yu. Pogromsky and dr. ir. N. Van de Wouw for providing me with this opportunity. Next to the people directly involved in the project I would like to thank those who contributed to this project by their support throughout the years. I would like to thank the following people in particular: My parents, Anne and Henri¨ette for your continuing love, support and believe in me. Martin and Anne Fleur for you are always there for me. My grandfather Jan Muntendam, for the inspiration that helped me choose mechanical engineering. Mieneke, for your helpfull efforts in the final phase of this thesis. And finally, Sonja, for everything. D.J. Rijlaarsdam, June 2, 2008

1 This work was partially supported by the Dutch-Russian program on interdisciplinary mathematics ’Dynamics and Control of Hybrid Mechanical Systems’ (NWO grant 047.017.018).

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Huygens Synchronization in Various Dynamical Systems. Experimental Results.

1 Introduction

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Introduction

This morning, you were most probably woken up by your alarm clock. Let’s assume for a moment, that there is no such thing as work or obligations. You are free to stay in bed for as long as you wish. You’ll probably sleep late and wake up a few hours later. But your body will tell you to wake up in the morning anyway. How does your body know that it is time to wake up? And why does it tell you to wake up at this specific moment of the day? The reason for this is that your body synchronizes itself to the rhythm of day and night as perceived by your senses. This simple example shows how synchronization takes place at least once a day in everybody’s life. Apart from synchronizing sleep rhythms, numerous examples of biological synchrony exist. Synchronizing dynamics, for example, have been recorded between neurons in the brain playing a role in the storage of memory. And, for example, synchronization between neurons plays an important role in the comprehension of these very words you are reading. Apart from natural synchronization, man made synchronization occurs frequently and all around us. Turning on a radio or using a mobile phone will initiate a sequence of synchronized processes, allowing the device to function correctly. Perhaps the most famous example of man made synchronization, is the construction of the ’leap year’, making sure that mans’ calendar stays in synch with the cosmic one. Although it is clear that synchronization is a rather common phenomenon, the mechanisms behind these synchronizing phenomena are far from straightforward. A better understanding of these mechanisms is required to understand - and possibly control - the related processes such as epileptic seizures or high end encrypted communication. Many theoretical and experimental studies have, therefore, been performed, leading to a better understanding of these phenomena. This thesis aims to add to this field of knowledge by supplying additional and new analytical, numerical and experimental results generated by using a new experimental set-up developed at the Eindhoven University of Technology. These findings are also presented at ENOC 2008 (Rijlaarsdam et al., 2008). The main research objective of the project described in this thesis is to: Investigate, using a new experimental set-up, the existence and stability of synchronization regimes in coupled oscillatory systems. This main research objective is split up into four sub-objectives aiming to: 1. Construct soft- and hardware that enables effective and accurate operation of the experimental set-up. 2. Develop a model which accurately describes the dynamics of the experimental set-up. 3. Develop a closed loop system that facilitates a wide range of experimental studies, using the experimental set-up as a platform to resemble different types of dynamical systems for demonstration and research purposes. 4. Obtain experimental results, identifying possible synchronization regimes within the coupled dynamical systems under consideration and compare these results to theoretical as well as numerical results. The thesis consists of two main parts. The first part focuses on the description of the experimental set-up. In Chapter 2 the concept of synchronization is discussed from a historical point of view, including a detailed discussion of one of the first recordings of synchronization by Chistiaan Huygens in 1665, and a literature overview is presented. Chapters 3 and 4 respectively describe the experimental set-up and the derivation and identification of a model of the set-up. Finally, Chapter 5 concludes the first part of the thesis by analyzing the stability of possible synchronization regimes within the set-up and presenting numerical and experimental results that support this analysis. The second part of this thesis describes different case studies of synchronizing dynamical systems. A so called ’masking dynamics’ approach is introduced that enables experiments to be conducted with a set of user specified dynamics rather than those inherit to the set-up. Chapter 6 explores this

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1 Introduction

approach in detail. In Chapter 7 this same ’masking dynamics’ approach is used to model a system of coupled Duffing oscillators. In Chapter 8 the same procedure is followed to imitate a system of rotating discs. Both chapters provide analytical, numerical and experimental results confirming the presence and stability of synchronization regimes within the systems. In Chapter 9 preliminary results are presented showing the current state of events in modeling the classical Huygens set-up (two pendula mounted on a common frame). For the Huygens set-up, no experimental results are available yet. Chapter 9, however, describes the theoretical analysis of this system and an extensive analysis of the observed problems and their possible solutions. Finally, in Chapter 10 conclusions and recommendations for further research are provided.

Nomenclature and Preliminaries This section introduces the notation standards that are used throughout this report. Mathematical Notation • Matrices and vectors are denoted by bold typeface. Matrices are denoted by capitals, while vectors are denoted by non-capital typeface. • The · symbol on top of any parameter denotes differentiation with respect to time, i.e. x˙ =  dx1 dx2  dxn T dx . dt = dt , dt , . . . , dt

Experimental and Simulation Results

• Blue and red lines in any plot correspond to the corresponding oscillator color in the set-up or the subscript number 1 and 2 in the models. Black lines correspond to the motion of the beam / frame or the subscript 3 in the model. • Measurements are depicted by bold lines, while simulation results are depicted by thin lines. • Unless specified otherwise the measurement frequency equals 1 [kHz] in experiments.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

2 History and Motivation

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Part I

Huygens 2008 The world is my country, science is my religion. [C. Huygens] In the first part of this report the project and the experimental set-up are introduced. Starting with a literature overview and providing a historical overview of synchronization in Chapter 2 the set-up is introduced in Chapter 3. Chapter 4 continues with the derivation and identification of a model for the systems’ dynamics. Finally, this part is concluded by Chapter 5 which introduces the notion and definition of synchronization and applies this definition to simulation and experimental results. Apart from showing the validity of the derived model it provides a kick-off for the second part of this report, which deals with experimental results in more detail.

2 2.1

History and Motivation A Brief History of Christiaan Huygens

In a time when the followers of Aristoteles could no longer stop the upcoming tide of the free thinking society, a time when science started to emerge in a form that relied on experiments rather than religion, Christiaan Huygens was born on April 14th 1629 (†July 8th 1695). This was the time when Galileo Galilei unraveled the mystery of kinematics and the heliocentric view gained ground over the ancient way of viewing humanity as the center of the universe. It was the age of great minds like Ren´e Decartes, Isaac Newton, Blaise Pascal, Henry Briggs and Pierre de Fermat. In this thrilling time, Huygens started his education in Leiden (The Netherlands), at age 16 (Andriesse, Figure 2.1: Christiaan Huygens by 1993). Pierre Bourguignon, 1688. (Andriesse, Originally Christiaan Huygens set out to study law, but 1993) his interest soon shifted towards mathematics and later physics. In mathematics he published ’De iis quae liquido supernatant’, about the laws that govern floating bodies and wrote his unpublished work ’De circuli magnitudine inventa’, about the circle perimeter. Among his greater works appeared ’De motu corporum ex percussione’ in 1652, which describes the motion of coliding objects and ’De vi cetrifuga’ in which Huygens investigates the centripetal force. Huygens had a practically motivated interest in the dynamics of pendula clocks. Galilei had already suggested the usage of such mechanism for timekeeping, but the realization of the pendulum clock came from Huygens. His fascination with pendulum clocks lead to discoveries that are used till this very day and were published in his masterpiece ’Horologium oscillatorium’ in 1673 (Huygens, 1934). During his research into the design of pendulum clocks, Huygens aimed to make the frequency of the pendulum independent of its amplitude. In his search for this isochronous motion he discovered the cyclo¨ıd path that the pendulum blob should follow in this case. Moreover, he aimed to find a better description for the pendulum than a massless rod with a point mass at its end. Therefore, he introduced the notion of the ’instantaneous point of zero velocity’ and in doing so he set the first Huygens Synchronization in Various Dynamical Systems. Experimental Results.

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2.2 300 Years of Synchronization

step in describing the kinematics of a rigid body. Finally, he set out to use the pendulum clock to provide a solution for the problem of finding the longitude coordinate at sea. This quest lead to the invention of the spring driven escapement mechanism that allows a pendulum clock to run. But, more importantly, it lead to his observation of synchronization (’sympathie des horloges’), which resulted in more than 300 years of research and ultimately laid the foundation for this thesis.

2.2

300 Years of Synchronization

In 1657 Salomon Coster built the first pendulum clock after Huygens’ design (Huygens, 1986). Due to the excellent properties of these clocks Huygens figured this could be the solution to the ’longitude problem’ put forward by The Royal Society of London. Intermezzo: The Longitude Problem The Royal Society of London presented the challenge to solve the problem of finding the longitude coordinate at sea. One solution for this problem is to use the time difference between the last departed harbor and the present location. In order to determine this time difference, one needs an accurate clock that provides the time at the reference location and use this find the time difference between the present location and the last departed harbor (for example at mid-day, when the sun is at its highest point in the sky). Since the total time difference around the globe is 24 hours and the total number of degrees longitude for a complete circle around the earth is 360◦ , the time difference can be converted to the number of degrees longitude East or West with respect to the last departed harbor.

The application of Huygens’ clock design for this purpose was investigated in cooperation with Alexander Bruce (2nd Earl of Kincardine) and tested from 1662 till 1665. In 1669 Huygens suggested the use of two pendulum clocks on a ship (Huygens, 1669), since one clock should be able to provide timekeeping while the second clock is being cleaned or repaired. During a time of illness Huygens was bound to stay in bed and two such pendulum clocks, mounted on a common frame were located in his bedroom (see Figure 2.2). When observing the motion of these clocks he perceived ’an odd kind of sympathy between these watches suspended by the side of the other ’. He observed that the motion of the clocks converged to an anti-phase synchronized state and reported his findings in 1665 (Huygens, 1932, 1983a,b).

Figure 2.2: A drawing of the set-up in which Christiaan Huygens observed synchronization (Huygens, 1932).

Although Huygens thought his findings added to the value of his design for timekeeping at sea (Yoder, 1990), the Royal Society thought otherwise (Birch, 1756) and discarded the use of pendulum clocks for solving the longitude problem.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

2.2 300 Years of Synchronization

17

We will not dwell on the solution of the longitude problem, since the invention of the global positioning system has solved this problem once and for all. Instead, attention is focussed on the synchronization phenomenon or ’sympathie des horloges’ that was observed by Huygens. This phenomenon has inspired many scientists over the next three centuries. In the beginning of the 20th century Korteweg studied Huygens- or frequency synchronization using normal mode analysis (Korteweg, 1906). He was the first to suggest that friction would make anti-phase synchronization the most favorable mode for Huygens system. But the escapement mechanism was disregarded in this analysis. In (Bennett et al., 2002) the authors include the escapement mechanism and study the influence of the mass ratio between the oscillators and the connecting frame. By studying a Poincar´e map of the systems’ trajectories they show that the limiting behavior will result in anti-phase synchronization, quasiperiodic motion or ’beating death’ (one of the oscillators comes to a standstill). The quasiperiodic motion is only observed for largely non-identical oscillators and ’beating death’ is caused by the escapement mechanism used. Therefore, their findings largely correspond to the original findings reported by Huygens. In (Pantaleone, 2002; Blekhman, 1988; Kuznetsov et al., 2007; Belykh et al., 2008) the authors provide an analysis of Huygens synchronization, using Van der Pol type oscillators. These studies report coexisting anti-phase and in-phase synchronization regimes. Pantaleone also reports experimental verification of the in-phase regime, but is able to reproduce Huygens original findings by adding damping to the system (see Kortewegs original suggestion). In (Oud et al., 2006) the authors present an extensive experimental analysis of the Huygens phenomenon and find both in- and anti-phase synchronization. A recent overview and analysis of Huygens synchronization is presented in (Senator, 2006). Apart from uncontrolled synchronization, studies of controlled synchronization are presented in (Pogromsky et al., 2003, 2006; Ananyevskiy et al., 2008). The authors present analysis of closed loop Huygens-like systems where control input is applied to either the oscillators or the connecting frame (Ananyevskiy et al., 2008). They show that by applying control, one can control both the type of synchronization that occurs and the final energy (amplitude) of the oscillators completely or at least to a certain extend (Ananyevskiy et al., 2008). The general interest in synchronization has increased significantly over time, since examples of this type of phenomena appear in a variety of fields, such as electrical engineering and biology. In electrical engineering the studies of Appleton (Appleton, 1922) laid the foundations for early radio communication. Moreover, the principles of synchronization emerge each time a mobile phone is used or a radio is tuned to a specific frequency. A well known report of biological synchronization is published in (Buck, 1988), where the author describes synchronization of rhythmic flashing of a large population of fireflies. In (Gray et al., 1989) the authors report measuring synchronization of neuronal activity in a cat brain and (Oud et al., 2004; Rijlaarsdam et al., 2007) investigate the mechanism behind such synchronization in more detail. In (Michaels et al., 1987) the authors report synchronization in population of pacemaker cells and (Strogatz and Stewart, 1993) investigates the role of synchronization in biology as well. Moreover, in (Strogatz, 2003) the role of synchronization in sleep rhythms is investigated. Summarizing, Huygens’ observation of synchronization of two pendulum clocks in the 17th century lead to an impressive number of studies and has been investigated from both analytical and experimental perspective. This thesis aims to add to this field by supplying additional analytical and experimental results, based on a novel set up that has been developed at the department of Mechanical Engineering of the Eindhoven University of Technology (Tillaart, 2006). This set-up allows to study both controlled and uncontrolled synchronization. Moreover, it is possible to modify this system to represent different types of dynamical systems and the systems’ parameters can easily be modified between and during experiments. Therefore, this set-up allows for the study of (Huygens) synchronization in an accurate and divers way.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

18

3

3 Set-up

Set-up

A novel set-up, designed to investigate Huygens-like synchronization phenomena, has been developed at the Eindhoven University of Technology (Tillaart, 2006). This set-up was built to continue and expand the efforts of Oud (Oud et al., 2006), who studied Huygens synchronization intensively in 2006, using a less sophisticated set-up. Therefore, the majority of the recommendations provided by Oud in his thesis are taken into account in the new design. The realization of the set-up was completed in May 2007 and the final result is depicted in Figure 3.1.

Figure 3.1: Photograph of the set-up.

3.1

Set-up Design

The set-up consists of two oscillators of mass mi , i = 1, 2, mounted on a common frame of mass m3 (see Figure 3.2) and is equipped with three actuators and position sensors on all degrees of freedom. The parameters of primary interest in this set-up are presented in Table 3.1 and will be discussed in more detail in Chapter 4. Furthermore, although the masses of the oscillators are fixed, the mass of the connecting beam (m3 ) may be varied by a factor 10. This allows for mechanical adjustment of the coupling strength. x3

x2

x1

κ3

κ1

β3

β1

F3

F1

κ2 β2

m1

m2

F2 m3

Figure 3.2: Schematic representation of the set-up at the Dynamics and Control laboratory (Dept. of Mechanical Engineering, Eindhoven University of Technology).

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

3.2 Technical Discussion of the Set-Up

19

Table 3.1: Parameters of interest in the experimental set-up. Oscillator 1 Oscillator 2 Frame / beam (3) Mass m1 m2 M Stiffness κ1 (·) κ2 (·) κ3 (·) Damping β1 (·) β2 (·) β3 (·) The set-up design, although closely related to Huygens original set-up (see Figure 2.2), is designed to allow easy modification of the systems dynamics by actuation and to provide a set of equations of motion that are as simple as possible, in order to simplify theoretical analysis. Before continuing with the identification of a model for this set-up and the analytical, numerical and experimental results, the introduction of the set-up will be completed by briefly discussing the most important technical issues related to this project.

3.2

Technical Discussion of the Set-Up

This section briefly introduces the practical implementation of the set-up. However, details are not provided in the main text. The reader is referred to Appendix C for more information and to (Tillaart, 2006) for details related to the design of the set-up. First of all, the choice of the Data Acquisition System (DACS) will be motivated and the safety system that has been designed to allow experiments without human presence will be discussed. Next, the issue of tuning the actuators and calibrating the sensors will be introduced. Finally, when performing experiments, damping and hysteresis proved to be an issue. The identification of these problems and their solution will be discussed in the last paragraph of this chapter. 3.2.1

Data Acquisition, Safety System and Software

After the initial assembly, the set-up already contained both the actuators and the sensors, but no data acquisition system was selected. A complete overview of the considered possibilities and their main advantages and drawbacks is provided in Appendix C.4. The choice for the TU/e’s MicroGiant system appeared to be the most suitable. Furthermore, no emergency brake system was present in the original design. In order to make sure the system will automatically shut down if excessive motion (resonance phenomena) occurs or if the power running through the actuators exceeds the maximum allowed value of 8W a safety system has been designed and realized. This system consists of a set of (mico-) switches and fuses that control both the motion and the heat production of the system. A diagram showing the configuration of the complete system of DACs, set-up, amplifiers and set-up as a whole is provided in Appendix C.1, while the details of the safety system are provided in Appendix C.2. Finally, software (see the accompanying CD) was designed to allow easy and completely autonomous operation of the set-up. The software allows for multiple experiments with different parameters and will automatically detect synchronization before switching to a new experiment. This feature allows for conducting large amounts of experiments, or very long tests without the need of human presence. 3.2.2

Tuning the Actuators and Sensor Calibration

The set-up consists of three position sensors (Linear Variable Differential Transformers) and three voice coil actuators. In Appendix C.7 the specifications of the sensors are supplied and the specifications of the actuators are supplied in Appendix C.8. The sensors are calibrated such that 1 [V ] ∼ 5 [mm]. However, the calibration of the actuators yielded that the actuator / amplifier combinations are significantly non-identical. Therefore, the difference in actuator response to an identical signal, has been identified. The details of this experiment are supplied in Appendix C.5.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

20

3.2 Technical Discussion of the Set-Up

The difference in actuator strength has been compensated by multiplying the outgoing signals with an appropriate gain. 3.2.3

Viscous Damping and Hysteresis

During the process of identifying the set-up viscous damping influences as well as hysteresis and stick-slip behavior have been identified. A thorough analysis of these effects is provided in Appendix C.6.1 (viscous damping influences) and C.6.2 (hysteresis and dry friction). Using analytical as well as experimental results the main cause for viscous-like friction has been identified to be the back emf in the voice coils. Furthermore, hysteresis has been identified, but these influences are small and will not be compensated for at this point. The location of the dry friction element has also been identified and will be avoided during experiments, since removal of this element proved to be too time consuming for now.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

4 Modeling the Set-Up

4

21

Modeling the Set-Up

In order to correctly design and develop experiments, an accurate model of the experimental set-up is necessary. In this chapter such model is derived and identified. A linear model is derived using the Lagrange formalism and written in the appropriate dimensionless form. Next, the parameters of the system are systematically identified using a specially designed input signal and Kalman filter. Finally, attention is paid to the role of nonlinearities present in the system.

4.1

Equations of Motion

In this section the equations of motion of the set-up are derived and written in dimensionless form. A schematic representation of the set-up is depicted in Figure 4.1. x3

x2

x1

κ3

κ1

β3

β1

u3

u1

κ2 β2

m1

m2

u2 m3

Figure 4.1: Schematic representation of the set-up.

Here mi ∈ R>0 , xi ∈ R, i = 1, 2, 3, are the masses and displacements of the oscillators and the beam respectively. Functions κi : R 7→ R, βi : R 7→ R describe the stiffness and damping in the system. Finally, ui are the actuator inputs. First of all, the equations of motion for the system depicted in Figure 4.1 will be derived using the Lagrange formalism. A detailed derivation is provided in Appendix A.1. Although this approach allows the derivation of the equations of motion for nonlinear springs κi (·) and damping βi (·), a linear model appears to be sufficiently accurate and therefore linear springs κi = ki and linear viscous damping βi = bi are assumed. Under these assumptions the equations of motion expressed in absolute coordinates x = [x1 x2 x3 ]T are: x ¨1

=

x ¨2

=

x ¨3

=

−ω12 ∆x1 − 2ζ1 ω1 ∆x˙ 1 + c1 u1 (t)

−ω22 ∆x2 − 2ζ2 ω2 ∆x˙ 2 2 X  µi ωi2 ∆xi + 2ζi ωi i=1

where ∆xi = xi − x3 and ωi =

q

(4.1)

+ c2 u2 (t)

(4.2)

 ˜3 (t), ∆x˙ i − ω32 x3 − 2ζ3 ω3 x˙ 3 + c3 u

(4.3)

ki −1 ], mi [rad s th

ζi =

bi 2mi ωi

[−] the undamped eigenfrequency and

mi the dimensionless damping of the i subsystem respectively. Furthermore, µi = m [−] is the 3 dimensionless coupling strength, u ˜3 (t) = u3 (t) − u2 (t) − u1 (t) [V ] the nett input signal to the beam and ci [m s−2 V −1 ] accounts for the amplifier / motor constants.

In order to write the system of equations (4.1) - (4.3) in dimensionless form, define the following set of dimensionless parameters:

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

22

4.2

System Identification

Table 4.1: Dimensionless parameters. τ∗ = ω ¯t ωi = ω ¯ ̟i∗ xi = ℓξi∗ ui (¯ ω t) = νvi∗ (τ ∗ ) λ∗i = (ν ω ¯ 2 /ℓ)ci Here ∗ indicates a dimensionless parameter (this notation will be omitted in the sequel to increase readability), ω ¯ = 12 (ω1 + ω2 ) [rad s−1 ] is the mean eigenfrequency of the oscillators, ℓ = 5 [mm] the stroke of the oscillators and ν = 0.42 [V ] the maximum input that can be applied to the actuators. In dimensionless form the system of equations (4.1) - (4.3) now becomes: ξ1′′ ξ2′′

= −̟12 ∆ξ1 − 2ζ1 ̟1 ∆ξ1′ + λ1 v1 (τ ) = −̟22 ∆ξ2 − 2ζ2 ̟2 ∆ξ2′ + λ2 v2 (τ )

ξ3′′

=

2 X i=1

(4.4) (4.5)

  µi ̟i2 ∆ξi + 2ζi ̟i ∆ξi′ − ̟32 ξ3 − 2ζ3 ̟3 ξ3′ + λ3 v3 (τ ),

(4.6)

where ∆ξi = ξi − ξ3 and ′ denotes differentiation with respect to the dimensionless time τ . The following part of this section will deal with the identification of the parameters present in the model and the experimental validation of this model.

4.2

System Identification

This section deals with the identification of the model (4.4) - (4.6) proposed in the previous paragraph. First, a suitable excitation signal for identification procedure will be designed. The identification of the model will then proceed in two main steps. First the individual subsystems will be treated as decoupled oscillators, each governed by the dynamics as described by corresponding equation in the complete model, but without the coupling and difference terms. The identification results are validated by comparison between the measured and model response and quantified by an error penalty function. After the identification of the nine oscillator-specific parameters the identification is completed by determining the two remaining coupling parameters. Finally, nonlinearities in the systems’ stiffness characteristics are identified for use in the second part of this report. These results are then compared to the linear approximation that is present in the identified models. 4.2.1

Design of the Excitation Signal

A very important part of identifying a dynamical system is the design of the excitation signal. A signal with sufficient frequency content and a maximal fraction of input power used for identification is to be designed. In order to quantify these statements the following signal properties are proposed: Definition 4.1 (Crest Factor (Pintelon and Schoukens, 2001)). The crest factor Cr(u) of a signal u(t) is given by the ratio between the (absolute) peak value upeak and its Root Mean Square (RMS) value urms in the frequency band of interest: Cr(u) =

max |u(t)| t∈T

ku(t)k2

,

(4.7)

where T is the set of relevant time instances. The crest factor is a measure for the compactness of an excitation signal. Definition 4.2 (Time Factor (Pintelon and Schoukens, 2001)). The time factor T f (u) of a signal u(t) is given by:   2 1 2 Urms T f (u) = max Cr (u(t)) , k∈F 2 |U (k)|

(4.8)

where F is the frequency band of interest and U (k) the frequency spectrum of the signal u(t) over this spectrum. The time factor provides an estimate of the minimal measurement time needed to obtain a good FRF measurement over a frequency band F using the input signal u(t). Huygens Synchronization in Various Dynamical Systems. Experimental Results.

4.2 System Identification

23

For the identification of the system a signal is selected with a frequency content between 0.01 Hz and 25 Hz, since previous experiments showed that the systems dynamics are located within this frequency band. The signal is constructed as a multisine with 25 sines, according to: v(t, φ) =

Nf X

Ak cos (2πfk t + φk )

(4.9)

k=1

with Nf = 25, fk linearly spaced between 0.01 Hz and 25 Hz and a flat spectrum Ak = 1. The phase shifts φk are selected such that the crest factor is minimized. This minimization is performed according to the infinity norm algorithm introduced in (Pintelon and Schoukens, 2001) which solves the following optimization problem:   (4.10) min lim kv(t, φ)k2p p→∞

φ

where the 2p-norm (p ∈ Z>0 ) is defined as: kv(t, φ)k2p



1 = T0

ZT0 0

1  2p

v 2p (t, φ) dt

,

(4.11)

and T0 is the period time of the signal v(t, φ). By minimizing the 2p-norm using a constraint minimization algorithm, the optimal phase shift vector φ = [φ1 , φ2 , . . . , φNf ]T , with respect to minimization of the crest factor is approximated. Increasing p results in a converging algorithm that provides a signal with the required frequency spectrum and a minimum crest factor. Such signal provides the most effective use of the input power for identification, as opposed to a signal containing large peaks. The optimization problem specified by (4.10) has been solved for the specified spectrum up to p = 10 and the resulting phase shifts are presented in Appendix B.2 (Table: B.2). This optimization has also been performed for the same spectrum but with Nf = 50. However, computational limits only allowed for convergence up to p = 2 within reasonable calculation time. Results are presented in Appendix B.2 (Table: B.3). A sample of the resulting signal v(t) for Nf = 25 is presented in Figure 4.2. 1

0.8

0.6

0.4

v(t) [−]

0.2

0

−0.2

−0.4

−0.6

−0.8

−1

0

2

4

6

8

10

12

τ [−] Figure 4.2: Optimized multisine, using the infinity norm algorithm.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

24

4.2

System Identification

The signal presented √ in Figure 4.2 has a crest factor of Cr(v(t)) = 1.4838 which is close to that of a single sine ( 2). The time factor equals approximately T f (v(t)) = 200 [s]. Therefore a measurement of length 200 s suffices to obtain a good measurement for all frequencies in the frequency band F = [0.01 25] [Hz]. 4.2.2

Identification of the Individual Subsystems

The model in (4.4) - (4.6) is validated in two steps. First of all, attention is focussed on the individual oscillators: blue (1), red (2) and the connecting frame / beam (3) separately. Then, after identifying the parameters for each oscillator, the coupling parameters are identified. In order to quantify how well the model corresponds to the measured dynamics the following penalty parameter is defined: Definition 4.3 (RMS Error Parameter (RmsEP)). Define a measurement ym (t) defined on an set of time instances t ∈ T ⊂ R≥0 and obtained by supplying the system with an input input wm (t). Furthermore, define a model M, that results in an output ys (t) when provided with the same input wm (t). Next define the error between the output of the model ys (t) and the measured signal ym (t) as e(t) = ym (t) − ys (t). The Root Mean Square (RMS) Error Parameter is defined as the ratio between the RMS value of the measured signal and that of the the error signal: ke(t)k2 (4.12) χ(wm ) = kym (t)k2

The RmsEP provides a measure for the effectiveness of the fit over the measured interval T , for a given input wm (t). The model for the individual oscillators is easily obtained from (4.4) - (4.6) and equals: 2 Mi (θi ) : ξ¨i = −θ1,i ξi + 2θ2,i θ1,i ξ˙i + θ3,i vi (t),

(4.13)

with θi = [̟i ζi λi ]T the parameter vector for the ith oscillator. The identification of the nine oscillator specific parameters is performed by exciting each oscillator with the signal v(t) (length 200 [s]) that was derived in the previous paragraph, while the position of the other two oscillators remains fixed. Next, an extended Kalman filter (Gelb, 1996) is used to estimate the parameters (three at a time). The (initial) parameters of the filter are chosen as follows: Covariance Matrices in the Kalman Filter Assuming that all uncertainties in the model are uncorrelated and the noise-distorted measured states are uncorrelated as well, both the measurement noise covariance R and the process noise covariance Q are diagonal matrices. Since, det(R) 6= 0 to avoid singularity of the algorithm the diagonal elements of R are chosen non-zero. These are chosen according to the a priori knowledge of the measurement uncertainties. These uncertainties are measured to be in the order or magnitude of 10−6 when the system is in rest. In order to allow for larger dynamic errors the elements of R are chosen Rj = 10−4 . The proces noise covariance matrix is divided on two parts. First of all, the part corresponding to the measured states is chosen as a block matrix which elements equal Qj = 10−4 . The remaining diagonal elements of Q are chosen 0 since the parameters are assumed to be constant in time. The initial covariance matrix P0 is also chosen as a diagonal matrix as well, since the parameters are assumed to be independent and states are assumed to be independent as well. The diagonal elements of P0 again represent two parts. The first block of P0 represents the initial error between the measured and estimated state. These elements can be small, since the initial conditions are well known. However, note that these elements should be non-zero, since P0 > 0 in order for the algorithm to function correctly. The second set of diagonal elements of P0 corresponds to the initial error between the actual and estimated parameters. These are unknown and therefore these elements should be chosen larger. Since experience shows that P0 adapts very quickly to the correct Huygens Synchronization in Various Dynamical Systems. Experimental Results.

4.2 System Identification

25

estimate during the iterative proces the diagonal elements are chosen as P0,j = 10−2 . The results of the Kalman filter provide a basis for manual fine-tuning. The necessity for manual fine-tuning is twofold. Firstly, the results provided by the Kalman filter clearly leave room for improvement. Secondly, the parameter estimates for the beam are derived from a measurement during which the red and blue oscillator were fixed to the beam. Therefore the mass of the beam is overestimated. Using the parameters obtained from the identification of the two oscillators (red and blue) this was corrected. After fine-tuning the parameters presented in Table 4.2 were obtained. Where ω ¯ = 21 (ω1 + ω2 ) = 13.2929 [rad s−1 ] and the related parameters for equations (4.1) - (4.3) are provided in Table B.1, in Appendix B.1. Table 4.2: Dimensionless parameters in equations (4.4) - (4.6). Oscillator 1 Oscillator 2 Frame / Beam (3) ̟i 0.9443 1.0557 0.7325 ζi 0.3362 0.4296 0.0409 λi 3.1054 · 105 3.4505 · 105 1.8686 · 104 A sample of the response of the blue oscillator and the beam to the input signal v(t) is compared to simulation results in Figure 4.3. A similar figure is provided for the red oscillator in Appendix B.4 (Figure B.2a). Furthermore, since the proposed model is linear, a step response provides good means of judging the degree up to which the proposed model models the systems’ dynamics. The measured response of the system to a step signal with increasing amplitude (to yield nonlinearities) is provided in Figure 4.4 and B.2b. 0.6

Measurement

1

Simulation

Measurement

Simulation

ξ3 [−]

ξ1 [−]

0.4 0.2 0 −0.2

0.5

0

−0.5

−0.4 −0.6 0

10

20

30

40

50

60

0

10

20

0

10

20

30

40

50

60

30

40

50

60

0.06

0.2

0.04

0.1

e3 [−]

e1 [−]

0.02 0 −0.02 −0.04 −0.06

0 −0.1 −0.2

0

10

20

30

τ [−] (a)

40

50

60

τ [−] (b)

Figure 4.3: Measured and simulated response (samples) to the input signal v(t). (a) Blue oscillator (b) Beam / frame, (Top) Response, (Bottom) Error: ei = ξi,s − ξi,m , where ξi,s denotes simulation results and ξi,m the corresponding measured response.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

26

4.2

1.5

1.5

Measurement

Simulation

Measurement

Simulation

1

ξ3 [−]

1

ξ1 [−]

System Identification

0.5 0 −0.5

0.5 0 −0.5 −1

−1 −1.5

−1.5 0

500

1000

1500

0

500

0

500

1000

1500

1000

1500

0.1 0.5

e3 [−]

e1 [−]

0 −0.1 −0.2

0

−0.5

−0.3 0

500

1000

1500

τ [−]

τ [−]

(a)

(b)

Figure 4.4: Measured and simulated response to a step signal with increasing amplitude. (a) Blue oscillator (b) Beam / frame, (Top) Response, (Bottom) Error: ei = ξi,s − ξi,m , where ξi,s denotes simulation results and ξi,m the corresponding measured response.

Both the step response experiment and the experiment in which the oscillators were excited with the signal v(t) yield that the proposed model and the identified parameters fit the measured response well. From these experiments it is observed that both the red and the blue oscillator respond in a linear manner while the beam / frame inhibits nonlinear behavior. In the step-response experiment this becomes clear from the fact that the simulated (linear) model fits the measured data worse as the amplitude increases. In order to quantify this observation the RmsEP has been calculated according to Definition 4.3. The results are provided in Table 4.3 and correspond the observation that the linear model, models the oscillators better than the beam / frame. The estimated parameters for the beam are however sufficient for our purposes and can easily be refined for specific use by re-estimating the parameters for a specific frequency band, amplitude and / or nonlinear fit. In Section 4.2.4 the nonlinearity in the stiffness of the beam / frame is identified for later purposes. Table 4.3: RMS Error Parameter for the three subsystems, where wf resembles the experiment with the multisine and ws refers to the step-response experiment. Oscillator 1 Oscillator 2 Frame / Beam (3) χ(wf ) 0.0649 0.1025 0.2267 χ(ws ) 0.0666 0.0628 0.2796 4.2.3

Coupling Parameters

In addition to the previously derived nine parameters for the three individual oscillators, two coupling parameters µ1 and µ2 remain unknown. Since nine out of eleven parameters are known the remaining two may be estimated using an extended Kalman filter (Gelb, 1996) applied to the complete, coupled model (4.4) - (4.6), using the same covariance matrices as for the individual oscillators.. The estimate has been performed on the free motion of the system, i.e. vi (t) = 0 because this way, parameter uncertainties in λi will not translate into increased uncertainty in the estimate of µi ∀ i = 1, 2. However, uncertainties in ωi and ζi ∀ 1 = 1, 2, 3 will inevitably influence the quality of the estimate of µi . The resulting parameters are presented in Table 4.4. Table 4.4: Estimated coupling parameters. µ1 = 0.0411 µ2 = 0.0578 Huygens Synchronization in Various Dynamical Systems. Experimental Results.

4.2 System Identification

27

e1 [−]

ξ1 [−]

Figure 4.5 shows the obtained measurement and the simulation results. As with the identification of the first nine parameters, the RmsEP has been calculated for this experiment to provide a measure of model accuracy. The obtained values for χ(0) are provided in Table 4.5. As becomes clear from these results, the fit yields the most accurate result for the blue oscillator and the beam. The red oscillator however also follows the predicted trajectory rather well, as can be seen in Figure 4.5. 0

−0.5

0.1 0.05 0 −0.05 −0.1

0

10

20

30

40

50

60

e2 [−]

ξ2 [−]

−1

0 −0.2 −0.4 −0.6 0

10

20

30

40

50

1

1.5

2

2.5

3

3.5

4

4.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0.1 0.05 0

60

0.1 0.05 0 −0.05 −0.1

0.5

−0.05

e3 [−]

ξ3 [−]

−0.8

0

0

10

20

30

τ [−] (a)

40

50

60

0.02 0 −0.02 −0.04

τ [−] (b)

Figure 4.5: Simulation and experimental response of the complete system (a)− measurement, − simulation (b) Error: ei = ξi,s − ξi,m , where ξi,s denotes simulation results and ξi,m the corresponding measured response.

Table 4.5: RMS Error Parameter for different subsystems. Oscillator 1 Oscillator 2 Frame / Beam (3) χ(0) 0.05448 0.1307 0.0869 4.2.4

Nonlinear Stiffness

In the second part of this report detailed knowledge concerning the stiffness present in the system is required. By measuring the displacement of each of the oscillators and the beam in response to a very slow (60 [s]) ramp signal the spring characteristics of the three subsystems are obtained. The results are depicted in Figure 4.6 in combination with force-displacement curves predicted by the model.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

28

4.3 Concluding Remarks

1

0.8

0.6

Blue oscillator (measurement) Red oscillator (measurement) Beam (measurement) Blue oscillator (model) Red oscillator (model) Beam (model)

0.4

v(t) [−]

0.2

0

−0.2

−0.4

−0.6

−0.8

−1 −1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

∆ξi [−] Figure 4.6: Spring characteristics for both oscillators and the beam. − Measured stiffness, − Linear approximation (used in the derived model).

As becomes clear from Figure 4.6 the blue and red oscillator inhibit very linear and nearly identical stiffness characteristics. The force-displacement curves predicted by the derived model are almost indistinguishable from the measurements in this case. The stiffness of the beam / frame however, is much more nonlinear, as may be expected from te construction. The beam is connected to an S-shaped spring that causes the observed nonlinearity. Such springs where originally also present in the oscillators, but these have been removed to improve the performance of the set-up. The estimated value for the model appears to provide a good average and will clearly serve as a good estimate. In order to quantify the nonlinearities in the stiffness the measured force-displacement curves have been fitted by a 5th order polynomial. The least square approximation of these stiffness functions is given by: 5 X ρij (∆ξi )j (4.14) ̺i (∆ξi ) = j=0

where, ∆ξi = ξi − ξ3 , i = 1, 2, ∆ξ3 = ξ3 and ̺i (∆ξi ) is the force-displacement characteristic belonging to the ith oscillator and the coefficients ρij are provided in Appendix B.3. These approximations will be used to adjust the stiffness characteristics in the second part of this report.

4.3

Concluding Remarks

The preceding chapter deals with the identification of a linear model for a system that is only linear to a certain extend. Therefore, it is not to be expected to find an ’exact’ match between the measurements and the model. The results presented in this chapter show that the proposed model with the identified parameters models the dynamics of the system very well. However, as shown in paragraph 4.2.4, nonlinearities in the stiffness of the system do exist. This model uncertainty may be coped with at a later stage, since (nonlinear) identification or parameter estimation for a specific use (amplitude / frequency band) can be performed easily. Within the scope of this assignment the obtained model models the system well enough, since the beam is only expected to move with small amplitude, such that the linear approximation is sufficient.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

4.3 Concluding Remarks

29

Another cause for deviating behavior of the experimental set-up may be nonlinear damping influences. Although the role of hysteresis and dry friction has been discussed briefly in Section 3.2.3 and these phenomena have been recognized and identified, they are not part of the model. Concluding, although nonlinearities in both damping and stiffness are present their influence is small, as shown by the fact that the systems’ dynamics may be modeled by a linear model up to a large extend. The derived linear model is thus judged to be a suitable model for the experimental set-up and will be used throughout the sequel to predict and analyze the systems’ dynamics.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

30

5

5 Analysis and Simulation Results

Analysis and Simulation Results

This chapter presents an analysis of the dynamical behavior of the set-up. More specifically, it deals with the possibility of synchronized dynamics between the oscillators. Based on the derived model the system’s limiting behavior is analyzed, using on a set of definitions that is introduced at the start of this chapter. Next, analytical results are compared to simulations and the influence of non-identical oscillator parameters is investigated. Finally, experimental results are presented and compared to the analytical and simulation results.

5.1

Synchronization: Introduction and Definition

Before continuing with the analytical, simulation and experimental results the notion of synchronization should be defined in more detail. Due to the large amount of phenomena that is gathered under the term ’synchronization’ it is often difficult to correctly define synchronization. In (Pikovsky et al., 2001) the authors introduce the concept of synchronization as: ’Synchronization is the adjustment of rhythms of oscillating objects due to their weak interaction.’ Although the above concept provides an insightful idea of synchronization a more rigorous definition is provided in (Blekhman et al., 1997): Definition 5.1 (Asymptotic Synchronization (Blekhman et al., 1997)). Given k systems with state ξi ∈ Xi and output yi ∈ Yi , i = 1, . . . , k and given ℓ functionals gj : Y1 × . . . × Yk × T 7→ R1 , where T is a set of common time instances for all k systems and Yi are the sets of all functions from T into the outputs Yi . Furthermore, defining a shift operator στ such that (στ y)(t) = y(t + τ ), call the solutions ξ1 (·), . . . , ξk (·) of systems Σ1 , . . . , Σk with initial conditions ξ10 , . . . , ξk0 are called asymptotically synchronized with respect to the functionals g1 , . . . , gℓ if: ∀ j = 1, . . . , ℓ (5.1) gj (στ1 y1 (·), . . . , στk yk (·), t) ≡ 0

is valid for t → ∞ and some στi ∈ T.

Since exact (asymptotic) synchronization (of whatever type) is not expected to be observed in real life systems, because of non-identical oscillator properties and disturbances, a definition of approximate (asymptotic) synchronization is provided. This allows effective handling of synchronization phenomena observed from measurements. Definition 5.2 (Approximate Asymptotic Synchronization (Blekhman et al., 1997)). Using the notations introduced in Definition 5.1, systems Σ1 , . . . , Σk are called approximately asymptotically synchronized with respect to to the functionals g1 , . . . , gℓ if for some sufficiently small ε > 0: |gj (στ1 y1 (·), . . . , στk yk (·), t)| 6 ε

∀ j = 1, 2, . . . , ℓ

(5.2)

is valid for t → ∞ and some στi ∈ T.

Finally, a specific type of synchronization, so called (approximate asymptotic) ’anti-phase’ synchronization’ will be of primary interest in the sequel. Using the preceding definitions of synchronization, a rigorous definition of anti-phase synchronization is provided. Again a definition is supplied that allows for (theoretical) anti-phase synchronization and more realistic, non-exact anti-phase synchronization. Definition 5.3 ((Approximate) Asymptotic Anti-phase Synchronization). Consider two systems Σ1 and Σ2 with initial conditions ξ10 and ξ20 and corresponding solutions ξ1 (ξ10 , t) and ξ2 (ξ20 , t). Furthermore, assume that both ξ1 (ξ10 , t) and ξ2 (ξ20 , t) are periodic in time with period T . The solutions of ξ1 (ξ10 , t) and ξ2 (ξ20 , t) is called (approximately) asymptotically synchronized in anti-phase if they are (approximately) asymptotically synchronized according to Definition 5.1 or 5.2, with: g(·) = ξ1 (·) − ασ( T ) ξ2 (·), (5.3) 2

with α ∈ R>0 a scale factor and σ( T ) a shift operator over half an oscillation period. 2 Huygens Synchronization in Various Dynamical Systems. Experimental Results.

5.2 Stability of the Synchronization Manifold

31

Remark. In case of anti-phase, equal amplitude oscillations, the definition of anti-phase synchronization presented in Definition 5.3 reduces to the condition that ξ1 +ξ2 → 0 as t → ∞ for ’complete’ asymptotic anti-phase synchronization. Correspondingly this yields the condition that ξ1 + ξ2 6 ε is valid for t → ∞ for approximate asymptotic anti-phase synchronization. In the sequel Definition 5.1 till 5.3 will be used to define (approximate) asymptotic (anti-phase) synchronization.

5.2

Stability of the Synchronization Manifold

Now that synchronization has been defined in a rigorous manner, attention is focussed on the stability of possible synchronization regimes in the dynamics of the set-up. In order to make sure that the following analysis is robust with respect to the presence of nonlinearities in the system, the notion of shape functions is introduced. The model (4.4) - (4.6) that has been derived, and identified contains linear approximations of the stiffness and damping present in the system. Although it has been shown that these approximations are sufficiently accurate to meet our needs, an analysis that takes into account the presence of possible nonlinearities is preferred. Therefore, consider the system of equations (5.4) - (5.6) which is identical to the model presented in equations (4.4) - (4.6), except for the introduction of the shape functions ηi (∆ξi ) i = 1, 2 η3 (ξ3 ) and σ3 (ξ3′ ), which allow for the introduction of nonlinearities in the stiffness and damping respectively. Note that the damping of the oscillators 1 and 2 may be altered in a similar way. However, these shape functions are omitted since only undamped oscillators are considered and such shape functions would vanish in the analysis. ξ1′′

=

ξ2′′

=

ξ3′′

=

−̟12 η1 (∆ξ1 ) ∆ξ1 − 2ζ1 ̟1 ∆ξ1′ + λ1 v1 (t)

−̟22 η2 (∆ξ2 ) ∆ξ2 − 2ζ2 ̟2 ∆ξ2′ + λ2 v2 (t) 2 X   µi ̟i2 ηi (∆ξi )∆ξi + 2ζi ̟i ∆ξi′ − ̟32 η3 (ξ3 ), ξ3 i=1

(5.4) (5.5) − 2ζ3 ̟3 σ3 (ξ3′ ) ξ3′ + λ3 v3 (t), (5.6)

For the system of (nonlinear) differential equations (5.4) - (5.6) the stability of the anti-phase synchronization manifold (as defined in Definition 5.3) is investigated using Lyapunov’s direct method. In order to proceed the oscillators are assumed to be undamped, i.e. ζ1 = ζ2 = 0 and no input is provided to the system, i.e. vi = 0 i = 1, 2, 3. Finally, define ηi (∆ξi ) i = 1, 2 and η3 (ξ3 ) to be odd, one to one, continuous shape functions, such that ∆ξi ηi (∆ξi ) has a zero only at ∆ξi = 0 or ξ3 = 0 and define a shape function σ(ξ3′ ), such that ξ3′ σ(ξ3′ ) > 0 ξ3′ 6= 0 and σ(0) = 0. The result of this analysis is provided in Theorem 5.1, while the complete analysis is provided in Appendix A.2. Theorem 5.1 (Global Asymptotic Stability of the Synchronization Manifold). Consider the system of nonlinear differential equations (5.4) - (5.6) and define the manifold S ⊆ R6 ˙ T ∈ R6 |ξ1 = −ξ2 , ξ ′ = −ξ ′ , ξ ′ = ξ3 = 0}. Next assume that the oscillators as the set S = {[ξ ξ] 3 1 2 are undamped and no input is provided to the system, i.e. 1. vi (t) = 0 ∀ i = 1, 2, 3 2. ζ1 = ζ2 = 0. Assume, furthermore, odd, one to one, continuous shape functions ηi (∆ξi ) ∀ i = 1, 2 and η3 (ξ3 ), such that ∆ξi ηi (∆ξi ) has a zero only at ∆ξi = 0 or ξ3 = 0 and a shape function σ(ξ3′ ) such that ξ3′ σ(ξ3′ ) > 0 ∀ ξ3′ 6= 0 and σ(0) = 0. Finally, assume the following oscillator properties: 1. η1 (·) = η2 (·) 2. ηi (·) such that

Rx 0

3. ̟1 = ̟2

s ηi (s) ds → ∞ if |x| → ∞ i = 1, 2, 3

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

32

5.3

Simulation Results

4. µ1 = µ2 . Then the system (5.4) - (5.6) will converge to S as t → ∞ for all initial conditions, Proof. The proof is provided in Appendix A.2 As shown in Theorem 5.1 the anti-phase synchronization manifold is globally asymptotically stable, where the anti-phase synchronization manifold defined in Theorem 5.1 corresponds to asymptotic anti-phase synchronization according to Definition 5.3, with α = 1. The following sections will present simulations and experimental results that show how this result translates into practical results.

5.3

Simulation Results

In this section the theoretical results from the preceding paragraphs are investigated in a simulation environment and using experimentally obtained results. Both the case of identical, undamped oscillators and the influence of non-identical oscillator properties are investigated. In order to gradually move from the theoretical model of identical, undamped, linear oscillators, towards the more realistic non-identical case, the section starts with simulations concerning identical oscillators. Next, attention is focussed on the case of non-identical, but still linear oscillators. Finally, the experimental results are presented, where nonlinear influences are small, but inevitably present. First of all, consider the system of equations (5.4) - (5.6) with linear, undamped oscillators and zero actuator input, i.e. ζ1 = ζ2 = 0, ηi (·) = σ(·) = 1 and vi = 0 ∀ i = 1, 2, 3. Furthermore, the parameters in the simulation are chosen as identified for the model in Chapter 4, except that ̟1 = ̟2 = 1 and µ1 = µ2 = 12 (µ1 + µ2 ) are identical for both oscillators. A simulation of such system, released from initial conditions ξ0 = [−1 − 0.8]T , is shown in Figure 5.1a. Although the system is released from an initial position close to the in-phase synchronized state, the trajectories of the system converge towards anti-phase synchronization. The system synchronizes according to Definition 5.3 (α = 0.9730 ε = 0.0243). Furthermore, the the phase difference between the oscillators, obtained by calculating the Hilbert transform of the measured signals, is shown in Figure 5.1b. After transient behavior this value converges to ∆φ1,2 → π as follows from Theorem 5.1.

ξi [−]

0.5

0

−0.5

−1

0

100

200

300

400

500

600

1

ξ3 [−]

0.5

∆φ1,2 (×π) [rad]

1

1.2

1

0.8

0.6

0.4

0

0.2 −0.5

−1

0

100

200

300

τ [−] (a)

400

500

600

0

0

100

200

300

400

500

600

700

800

τ [−] (b)

Figure 5.1: Synchronization between two identical undamped oscillators (simulation results). The system is released close to the in-phase synchronized mode (ξ0 = [−1 − 0.8]T ) and converges to the anti-phase synchronized (stable) mode. (a) Transient behavior. (Top: red and blue oscillator, Bottom: beam) (b) Phase difference between the oscillators.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

5.3 Simulation Results

33

Figures 5.2 and 5.3 show simulation results for the same situation as the previous example. However, instead of identical oscillators, non-identical oscillators have been used, i.e. ̟1 6= ̟2 and µ1 6= µ2 . All parameters are chosen as identified in Chapter 4, except that the oscillators remain undamped, i.e. ζ1 = ζ2 = 0. Figure 5.2 shows such simulation without energy input from the actuators. As one might expect the motion of all subsystems dies out eventually, since no stable mode of anti-phase synchronization can be sustained. In this case the synchronization manifold reduces to a single point ξ = 0. 1

ξi [−]

0.5

0

−0.5

−1

0

200

400

0

200

400

600

800

1000

1200

600

800

1000

1200

1

ξ3 [−]

0.5

0

−0.5

−1

τ [−] Figure 5.2: Synchronization between non-identical oscillators. No energy is supplied by the actuators. Top: red and blue oscillator, Bottom: beam.

By supplying energy (velocity dependent feed forward) to the oscillators to compensate for the energy dissipation due to the non-zero limiting motion of the beam, the result presented in Figure 5.3 is obtained. It follows that in this case the limiting behavior of the system differs from the anti-phase synchronized mode that was observed for identical oscillators. First of all, non-identical oscillator properties result in a limiting phase difference ∆φ1,2 = 0.91π. Furthermore, the steady state amplitudes of the oscillators differ significantly. In terms of Theorem 5.3 approximate asymptotic anti-phase synchronization is however still observed (α = 2.9345 ε = 0.1544).

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

34

5.4 Experimental Validation

1

ξi [−]

0

−0.5

−1

0

50

100

150

200

250

300

350

400

450

500

ξ3 [−]

1

∆φ1,2 (×π) [rad]

1.8

0.5

1.6

1.4

1.2

1

0.8

0.5

0.6

0

0.4

−0.5

0.2

−1

0

50

100

150

200

250

300

350

400

τ [−] (a)

450

500

0

0

200

400

600

800

1000

τ [−] (b)

Figure 5.3: Synchronization between non identical undamped oscillators (simulation results). The system is released close to the in-phase synchronized mode (ξ0 = [−1 −0.8]T ) and energy is supplied by the actuators to overcome the effects of non-identical oscillators (a) Transient behavior (Top: red and blue oscillator, Bottom: beam) (b) Phase difference between the oscillators.

5.4

Experimental Validation

In the preceding analysis and simulation results it has been shown that it is to be expected that anti-phase synchronization between the two oscillators in the experimental set-up wil be observed if no external force, other than damping compensation, is applied to the subsystems. To resemble the undamped oscillators used in the analysis and simulations as close as possible, a velocity depended feed forward is applied canceling the 2ζi ̟i ∆ξ˙i ∀ i = 1, 2 term in the equations of motion of the oscillators. Note that since the actual damping in the system is not linear viscous and the oscillators are non-identical the only way to achieve a sustainable mode of motion is to slightly overcompensate the damping. The results of such experiment with initial condition ξ0 = [−1 − 0.8]T are presented in Figure 5.4.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

35

1

1.8

0.5

1.6

0

−0.5

−1

0

50

100

150

200

250

1

1.4

1.2

1

0.8

0.6

0.5

ξ3 [−]

∆φ1,2 (×π) [rad]

ξi [−]

5.5 Concluding Remarks

0

0.4

−0.5

0.2

−1

0

50

100

150

τ [−] (a)

200

250

0

0

50

100

150

200

250

300

350

400

450

τ [−] (b)

Figure 5.4: Synchronization between two non identical oscillators (experimental results). Damping in the oscillators is (over-) compensated by means of velocity dependent feed forward. The system is released close to the in-phase synchronized mode (ξ0 = [−1 − 0.8]T ) and converges to the anti-phase synchronized (stable) mode. (a) Transient behavior (Top: red and blue oscillator, Bottom: beam) (b) Phase difference between the oscillators.

The experimental results closely resemble the situation depicted in Figure 5.3 (except for the presence of nonlinearities in the set-up). The limiting phase difference follows from Figure 5.4b and equals approximately ∆φ1,2 ≈ 1.065π. Furthermore, the steady state amplitude difference that was observed in the simulations is also obtained in the experiment. However, the experiment shows that approximate asymptotic anti-phase synchronization according to Definition 5.3 (α = 1.2927 ε = 0.1241) is indeed achieved. Note that the amplitude difference predicted by simulations (Figure 5.3) is about 2.3 times larger than observed in experiments. However, the measured and simulated phase difference only differ 2.5 % with respect to exact anti-phase synchronization or ∆φ12 = π

5.5

Concluding Remarks

In this section the definition of synchronization has been introduced. Global asymptotic stability of the synchronization manifold in case of identical oscillators has been proven and shown in simulations. Furthermore, the more realistic situation of non-identical oscillators has been investigated from both simulation and experimental perspective. The simulations and experimental results agree to a large extend and both indicate that approximate anti-phase synchronization (Definition 5.3) is achieved. This concludes the first part of this report. In the preceding part the set-up was introduced and both the modeling and analysis of the set-up have been presented. Attention has been focussed on identifying a correct (linear) model of the set-up and discussing the degree up to which such approximating model models the actual nonlinear system. Next, the properties of the derived model were analyzed with respect to synchronizing behavior and supported this analysis by both simulations and experimental arguments. The next part of this report will discuss and analyze more complex examples of synchronizing dynamical systems. Again analytical, simulation and experimental results are provided.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

36

Part II

Experimental Results Theory guides. Experiment decides. [Multiple authors] In the previous part, experimental synchronization has been observed between the system of coupled oscillators (4.4) - (4.6). These equations are repeated here for the readers convenience: ξ1′′ ξ2′′ ξ3′′

= −̟12 ∆ξ1 − 2ζ1 ̟1 ∆ξ1′ + λ1 v1 (τ )

= −̟22 ∆ξ2 − 2ζ2 ̟2 ∆ξ2′ + λ2 v2 (τ ) 2 X   µi ̟i2 ∆ξi + 2ζi ̟i ∆ξi′ − ̟32 ξ3 − 2ζ3 ̟3 ξ3′ + λ3 v3 (τ ). = i=1

The observed synchronizing behaviour in this dynamical system is in accordance with analytical results and simulations. This part continues the investigation of synchronization phenomena by using the potential of the experimental set-up to experimentally analyze synchronizing behaviour in different kinds of dynamical systems. Chapter 6 introduces the ’masking dynamics’ approach that is used in the sequel to modify the dynamics of the set-up. In Chapter 7 and 8 examples concerning a system of coupled Duffing oscillators and a system of rotating discs are discussed. Analytical and simulation results as well as experimental results are presented. Finally, Chapter 9 presents preliminary results for the case where the set-up is modified to model the classical Huygens set-up.

6

Masking of System Dynamics

In order to experimentally investigate synchronization phenomena in other types of dynamical systems than the experimental set-up, the notion of virtual dynamics is introduced. The virtual dynamics are the dynamics of the closed loop system consisting of the experimental set-up and a feedback controller that will be designed in the sequel. w(ξ, ξ ′ , τ ) = wM (ξ, ξ ′ ) + we (τ )

s(τ )

+ +

+ +

v ξ ′′ = f (ξ, ξ ′ , v)

ξ, ξ ′ + +

ℓ ξ ′′ = L(ξ, ξ ′ )

d

G

T−1

η ′′ = D(η, η ′ )

T

Figure 6.1: Schematic depiction of the feedback loop.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

6 Masking of System Dynamics

37

Consider the system depicted in Figure 6.1. Here, ξ ′′ = f (ξ, ξ ′ , v) are the dynamics of the set-up as approximated by the model (4.4) - (4.6) with input v and nonlinear stiffness characteristics (4.14). Next, s(τ ) represents input disturbances and the disturbance w(ξ, ξ ′ , τ ) represents both the model uncertainties wM (ξ, ξ ′ ) and output disturbances we (τ ). Furthermore, ξ ′′ = L(ξ, ξ ′ ) and η ′′ = D(η, η ′ ) are control dynamics that are yet to be defined. The operators T and T−1 allow for a coordinate transformation from and to the coordinate system of the experimental set-up to a new set of coordinates η in which the virtual dynamics are specified. The gain G scales the feedback with respect to the mass and motor constants in the set-up. Next, the control goal is formulated as follows: Definition 6.1 (Control Goal). Design a controller such that the closed loop dynamics of the system depicted in Figure 6.1 match a set of user specified open loop dynamics η ′′ = F(η, η ′ ). In order to fulfill the control goal, specified in Definition 6.1, the following relations are specified with respect to the feedback loop in Figure 6.1: L(ξ, ξ ′ ) = ′

D(η, η ) = G =

−f (ξ, ξ ′ , v = 0) Λ−1 ′

F(η, η ) Λ−1 ,

(6.1) (6.2) (6.3)

where Λ = diag[λ1 λ2 λ3 ] contains the gains from (4.4) - (4.6). The block ξ ′′ = L(ξ, ξ ′ ) = −f (ξ, ξ ′ , v = 0) Λ−1 cancels the set-ups’ dynamics ξ ′′ = f (ξ, ξ ′ , v). The resulting closed loop dynamics of the inner loop in Figure 6.1 (assuming s(·) = w(·) = 0) therefore equal ξ ′′ = Λ d. The block η ′′ = D(η, η ′ ) = F(η, η ′ ), finally, assures that the closed loop dynamics of the complete system equal those of the user specified open loop dynamics η ′′ = F(η, η ′ ). The process of masking the actual dynamics ξ ′′ = f (ξ, η ′ ) with a set of new dynamics η ′′ = F(η, η ′ ), by means of state feedback, leads to a virtual system that allows for a wide range of synchronization experiments. The results presented in the preceding paragraphs will be used in the remaining part of this report to model a number of different dynamical systems. However, before continuing with these results, please note the following potential drawbacks of the masking approach: Disturbance influences Although this project does not deal with disturbances in particular the role of model uncertainties and input and output noise should be considered. These disturbances influence the closed loop dynamics in two distinct ways. First of all, the feedback linearizing loop (inner loop) will not cancel dynamics that are not captured by the model, resulting in an unknown, unwanted part of the closed loop virtual dynamics. Furthermore, external noise will influence the amount up to which the desired dynamics are mimicked. Limited Actuator Power As in any controlled system the amount up to which the control goal is achieved depends (among other aspects) on the speed of calculation, measurement frequency and the actuator power that is present in the system. Especially the limited actuator power is expected to be a bottleneck since it directly influences the acceleration of the system and as such determines the fasted time scale in the virtual dynamics. It limits both the type of dynamical system that may be modeled and the parameter range that can be explored.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

38

7 Coupled Duffing Oscillators

7

Coupled Duffing Oscillators

This chapter presents analytical and numerical results as well as experimental results concerning synchronization between a pair of coupled Duffing oscillators. After the introduction of the dynamical system and the derivation of the equations of motion, a controller is proposed to allow experimental investigation of possible synchronization phenomena in the system. Next, attention is focussed on the stability of possible synchronization regimes in the set up. Finally, both numerical and experimental results are presented to investigate the existence of synchronization regimes from an experimental point of view.

7.1

Equations of Motion and Stability Analysis x3

x2

x1

k3

κd,1

b3

κd,2

m1

m2

m3

Figure 7.1: Schematic representation of the set-up modeling two coupled Duffing oscillators. Consider the dynamical system depicted in Figure 7.1, where xi ∈ R, mi ∈ R>0 and κd : R 7→ R is designed to resemble the cubic stiffness profile of the classical Duffing oscillator. For the sake of clarity some of the previously used notation is used in a new context in the remaining sections of this report. κd,i (∆xi ) = ωi2 ∆xi + βi ∆x3i (7.1) mi q q kβ,i ki , β = where ∆xi = xi − x3 , ωi = i mi mi and ki , kβ,i are coefficients of the cubic stiffness profiles that characterize the Duffing nonlinearity. The equations of motion of the system depicted in Figure 7.1 are: x ¨1 x ¨2 x ¨3

= = =

−ω12 ∆ x1 − β1 ∆ x31 −ω22 ∆ x2 − β2 ∆ x32

3 X i=1

  µi ωi2 ∆ xi + βi ∆ x3i − ω32 x3 − 2ζ3 ω3 x˙ 3 ,

(7.2) (7.3) (7.4)

mi where ζ3 = 2ωb33m3 is the dimensionless damping of the beam and µi = m the mass ratio between 3 the oscillators and the beam. In order to write the system (7.2) - (7.4) in dimensionless form, the following set of dimensionless parameters is introduced.

Table 7.1: Dimensionless parameters, duffing oscillators. τ∗ = ω ¯t ωi = ω ¯ ̟i∗
ω ¯ 2 ∗ ∗ x = ℓξ β= ℓ ϑ

Here ∗ indicates a dimensionless parameter and will be omitted in the sequel for the sake of readability, ω ¯ = 12 (ω1 + ω2 ) and ℓ = 5 mm. The system of equations in dimensionless form now

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

7.2 Simulation Results

39

becomes: ξ1′′ ξ2′′

= −̟12 ∆ ξ1 − ϑ1 ∆ ξ13 = −̟22 ∆ ξ2 − ϑ2 ∆ ξ23

ξ3′′

=

3 X i=1

(7.5) (7.6)

  µi ̟i2 ∆ ξi + ϑi ∆ ξi3 − ̟32 ξ3 − 2ζ3 ̟3 ξ3′ ,

(7.7)

where ′ indicates differentiation with respect to the dimensionless time τ . The possibilities of synchronizing dynamics in system (7.5) - (7.7) is first investigated from an analytical point of view. Using Lyapunovs direct method and LaSalles invariance principle, it is shown that anti-phase synchronization as defined by Definition 5.3 is globally asymptotically stable. These results are formalized in Theorem 7.1. Theorem 7.1 (Global Asymptotic Stability of the Synchronization Manifold: Duffing Oscillators). Consider the system of nonlinear differential equations (7.5) - (7.7) and define the manifold S ⊆ R6 as the set S = {[ξ ξ ′ ]T ∈ R6 |ξ1 = −ξ2 , ξ1′ = −ξ2′ , ξ3′ = ξ3 = 0}. Assume, furthermore, identical oscillator properties: 1. ̟1 = ̟2 . 2. ϑ1 = ϑ2 . 3. µ1 = µ2 Then the system (7.5) - (7.7) will converge to S as t → ∞ for all initial conditions. Proof. The proof provided in Appendix A.2 applies with ηi (qi ) = 1 + σ(q˙3 ) = 1.

7.2

βi 2 q , ωi2 i

i = 1, 2 and η3 (q3 ) =

Simulation Results

Consider the dynamics prescribed by (7.5) - (7.7) with the set of parameters presented in Table 7.2 (provided in both non-dimensionless and dimensionless form for convenience). Table 7.2: Simulation parameters, Duffing oscillators. (µ1,exp = 0.0411 [−] and µ2,exp = 0.0578 [−] as in the original set-up.) Non Dimensionless Dimensionless ω1 = ω2 = 18.3157 [rad s−1 ] ̟1 = ̟2 = 1 ϑ1 = ϑ2 = 7.4524 · 10−7 β1 = β2 = 4.4728 · 106 [rad s−2 m−2 ] 1 µ1 = µ2 = 2 (µi,exp + µ2,exp ) = 0.0495 [−] µ1 = µ2 = 0.0495 Figure 7.2a presents an example of the time response of the system of Duffing coupled oscillators and Figure 7.2b presents the phase difference ∆φ1,2 = (|φ1 − φ2 | mod 2π) as a function of time. As becomes clear from these results, the system gradually moves from the approximate in-phase synchronized initial condition towards anti-phase synchronized motion. Note that the apparent jump results from the fact that ∆φ1,2 is defined on a compact manifold ∆φ1,2 ∈ [0 2π]. After approximately τ ≈ 2500 [−] the system is in the (approximate) synchronized state as predicted by Theorem 7.1.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

40

7.3 Experimental Results and Controller Design

0.6

1.4

0.2 0 −0.2 −0.4 0

500

1000

1500

2000

0.1

∆φ1,2 (×π) [rad]

ξ1 , ξ2 [−]

0.4

1.2

1

0.8

0.6

ξ3 [−]

0.05

0.4 0

0.2 −0.05

−0.1

0

500

1000

1500

2000

0

0

500

1000

1500

2000

τ [−]

τ [−] (a)

(b)

Figure 7.2: Synchronization between two Duffing oscillators (simulation results) ξ0 = ′ ′ ′ T [ξ10 ξ10 ξ20 ξ20 ξ30 ξ30 ] = [−0.6 0 − 0.5 0 0 0]T (a) Transient behavior (Top: red and blue oscillator, Bottom: beam) (b) Phase difference between the oscillators.

7.3

Experimental Results and Controller Design

Experimental verification of the preceding results may be obtained by applying the ’masking dynamics’ approach introduced in Chapter 6. After the controller identities are defined, an example of such experiment is presented that relates to the previously discussed simulation results. Controller Design In order to assure that the dynamics of the virtual closed loop system mimic those described by (7.5) - (7.7) the following relations are defined in the masking approach that was introduced in chapter 6: LD (ξ, ξ ′ ) = −f (ξ, ξ ′ , v = 0) Λ−1 ′

′

DD (η, η ) = F(η, η ) G D = Λ−1 ,

(7.8) (7.9) (7.10)

where Λ = diag[λ1 λ2 λ3 ] contains the gains from (4.4) - (4.6). The block ξ ′′ = LD (ξ, ξ ′ ) = −f (ξ, ξ ′ , v = 0) Λ−1 cancels the set-ups’ dynamics ξ ′′ = f (ξ, ξ ′ , v). The block η ′′ = DD (η, η ′ ) = FD (η, η ′ ), furthermore, assures that the closed loop dynamics of the complete system equal those of the user specified open loop dynamics η ′′ = FD (η, η ′ ) which are specified by (7.5) - (7.7). Finally the virtual coordinate system η = [ξ1 ξ2 ξ3 ]T is equal that of the original set up, which yields a transformation matrix TD = I, where I is the identity matrix of appropriate dimensions.

Experimental Results In order to experimentally investigate synchronization behaviour in the system of coupled Duffing oscillators the controller identities that were designed in the previous paragraph are implemented and experiments are conducted using the parameter set provided in Table 7.3.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

7.4 Concluding Remarks

41

Table 7.3: Experimental parameters, Duffing oscillators. Non Dimensionless Dimensionless ω1 = ω2 = 18.3157 [rad s−1 ] ̟1 = ̟2 = 1 β1 = β2 = 4.4728 · 106 [rad s−2 m−2 ] ϑ1 = ϑ2 = 7.4524 · 10−7 µ1 = µ1,exp = 0.0411 [−] µ1 = µ1,exp = 0.0411 µ2 = µ2,exp = 0.0578 [−] µ2 = µ2,exp = 0.0578 Figure 7.3 shows that anti-phase synchronization, according to Definition 5.3 (α = 2.2200, ε = 5 · 10−4 ), does indeed occur in the experiment. After τ ≈ 600 [−] the system has approximately synchronized. However, the experimental results differ from the analysis and the simulations in two important ways. First, the final approximate anti-phase synchronized state is reached much faster in the experiments than in the simulations. A possible explanation for this is that (external) disturbances push the system away from the approximate in-phase synchronized state in the experiment, thus resulting in a faster convergence than observed in the simulations. Secondly, the final amplitude of the oscillators is not equal and the beam does not come to a complete standstill. In order to understand this effect note that analysis shows that for non-identical oscillators the synchronization manifold reduces to a single point (the origin). In order to retain a non-zero steady state response, external energy has to be added to the system. Furthermore, non-exact compensation of the friction in the system also causes a non-zero nett energy input to the system. Combining these two effects with the non-identical nature of the oscillators in the set-up leads to unequal amplitude oscillations and a remaining oscillation of the beam. This remaining motion of the beam is necessary to dissipate the energy overflow which in turn is required to sustain a non-zero steady state response. 0.6

1.8

0.2 0 −0.2 −0.4 0

100

200

300

400

500

600

ξ3 [−]

0.2

∆φ1,2 (×π) [rad]

ξ1 , ξ2 [−]

0.4

1.6

1.4

1.2

1

0.8

0.1

0.6

0

0.4

−0.1

0.2

−0.2

0

100

200

300

400

τ [−] (a)

500

600

0

0

100

200

300

400

500

600

τ [−] (b)

Figure 7.3: Synchronization between two Duffing oscillators (experimental results) ξ0 = ′ ′ ′ T [ξ10 ξ10 ξ20 ξ20 ξ30 ξ30 ] = [−0.6 0 − 0.5 0 0 0]T (a) Transient behavior (Top: red and blue oscillator, Bottom: beam) (b) Phase difference between the oscillators.

7.4

Concluding Remarks

It is shown that in a system of Huygens-like coupled Duffing oscillators the anti-phase synchronized mode is globally asymptotically stable. Analytical as well as simulation results are presented and the ’masking dynamics’ approach is utilized to experimentally investigate this phenomenon. This approach allows the alteration of the set-up dynamics to model the system of coupled Duffing oscillators. The experimental results support the results found in the preceding analysis and Huygens Synchronization in Various Dynamical Systems. Experimental Results.

42

7.4 Concluding Remarks

simulations. However, although the systems steady state response in the experiments is very close to anti-phase synchronized motion, the oscillators have different steady state amplitudes and the beam does not come to a complete standstill. The explanation for these observation is found in the non-identical oscillators in the experimental set-up.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

8 Coupled Rotating Discs

8

43

Coupled Rotating Discs

After the analysis of the system of coupled Duffing oscillators in the previous chapter this chapter aims to analyze synchronization phenomena in a system of coupled rotating discs. Starting with the derivation of the equations of motion of such system, the chapter continues with the design of the appropriate controller to guarantee the required closed loop dynamics in order to allow for experiments. Apart from the features that have been explored in the previous chapter, the current modulation shows the potential of the ’masking dynamics’ approach to allow the virtual dynamics to be defined in a different coordinate system than the dynamics of the set-up. With the definition of the dynamics as well as the appropriate controller completed, the chapter splits in two main parts. First, a simplified system is discussed. The stability of the anti-phase synchronization manifold is discussed from analytical as well as from simulation / experimental perspective. Moreover, the convergence rate of the system towards the stable anti-phase mode is discussed by supplying analytical as well as numerical arguments. Finally, the last part of this chapter supplies an analysis of the complete system. Again analytical, numerical and experimental results are supplied, investigating the existence of possible synchronization regimes in the system.

8.1

Equations of Motion and Controller Design g

θ3 θ1

θ2 k2

k1 1

ℓ1

J1

3

J3

2

ℓ3

me,1

J2 ℓ2 me,2

k3

me,3

b Figure 8.1: Schematic representation of the set-up modeling two coupled rotating elements. Consider the dynamical system schematically depicted in Figure 8.1 where θi ∈ S1 are the rotation angles, with respect to the world, me,i ∈ R>0 possible eccentric masses and Ji ∈ R>0 the moments of inertia of the discs. The equations of motions of the system under consideration are: θ¨1 θ¨2

2 sin(θ1 ) = −ω12 ∆ θ1 − ωe,1

(8.1)

=

(8.2)

θ¨3

=

−ω22 ∆ θ2 2 X i=1

−

2 ωe,2

sin(θ2 )

  2 sin(θ3 ), µi ωi2 ∆ θi − ω32 θ3 − 2ζ3 ω3 θ˙3 − ωe,3

q q me,i gℓi where ∆θi = θi − θ3 , ωi = Jkii , ωe,i = , ζ3 = Ji the inertia of the disc / eccentric mass combination.

b3 2ω3 Ji

and µi =

Ji J3 ,

(8.3)

where Ji = me,i ℓi + Ji

In order to write the system of equations (8.1) - (8.3) in dimensionless form, the following set of dimensionless parameters is defined: Huygens Synchronization in Various Dynamical Systems. Experimental Results.

44

8.2 Rotating Discs without Eccentric Masses

Table 8.1: Dimensionless parameters, rotating discs. τ∗ = ω ¯t ωi = ω ¯ ̟i∗ ωe,i = ω ¯ νi∗ Here, ω ¯ = 12 (ω1 + ω2 ) and ∗ indicates a dimensionless parameter and will be omitted in the sequel for the sake of readability. The resulting dimensionless equations of motion are: θ1′′ θ2′′

= −̟12 ∆ θ1 − ν12 sin(θ1 ) = −̟22 ∆ θ2 − ν22 sin(θ2 )

θ3′′

=

2 X i=1

  µi ̟i2 ∆ θi − ̟32 θ3 − 2ζ3 ̟3 θ3′ − ν32 sin(θ3 ),

(8.4) (8.5) (8.6)

where ′ indicates differentiation with respect to the dimensionless time τ . Controller Design Realizing a virtual dynamical system that mimics the dynamics prescribed by (8.4) - (8.6) requires the application of the masking approach discussed in Chapter 6 with the following identities: LR (ξ, ξ ′ ) = DR (η, η ′ ) = GR

=

−f (ξ, ξ ′ , v = 0) Λ−1 F(η, η ′ )

(8.7) (8.8)

Λ−1 ,

(8.9)

where Λ = diag[λ1 λ2 λ3 ] contains the gains from (4.4) - (4.6). The block ξ ′′ = LR (ξ, ξ ′ ) = −f (ξ, ξ ′ , v = 0) Λ−1 cancels the set-ups’ dynamics ξ ′′ = f (ξ, ξ ′ , v). The block η ′′ = DR (η, η ′ ) = FR (η, η ′ ), furthermore, assures that the closed loop dynamics of the complete system equal those of the user specified open loop dynamics η ′′ = FR (η, η ′ ) which are specified by (8.4) - (8.6). The virtual coordinate system η = [θ1 θ2 θ3 ]T is then obtained by choosing the appropriate coordinate transformation TR = diag[θm θm θm,3 ]. Note that θm and θm,3 are the angles of rotation corresponding to the strokes of the oscillators and the beam respectively. Although θm and θm,3 may be chosen arbitrarily, θm = rsi i = 1, 2 and θm,3 = rs3 are used in the sequel, where s is the stroke of the oscillators / beam and ri the radius of the corresponding disc. This assures that the distance traveled by the oscillators and beam in the set-up equals the arc traveled by a point on the outer perimeter of the corresponding disc.

8.2

Rotating Discs without Eccentric Masses

With the dynamics and the appropriate controller relations identified a slightly simplified version of the dynamics (8.4) - (8.6) is discussed first. Consider the system (8.4) - (8.6) with νi = 0 i = 1, 2, 3, i.e. in absence of the eccentric masses me,i . After analyzing the stability of possible synchronization regimes within this system both numerical and experimental results are presented that show the presence of synchronizing dynamics in the system. Finally, the convergence rate of the system dynamics towards the synchronization manifold is discussed in more detail. Since the system (8.4) - (8.6) with νi = 0 i = 1, 2, 3 is linear, one might originally aim to approach the study of possible synchronization regimes using a coordinate transformation to a system describing the dynamics of the sum of θ1 and θ2 . An analysis of the eigenvalues of such system should yield the presence of possible (anti-phase) synchronization regimes. Although this approach is utilized in the last paragraph to investigate the convergence rate of the system towards the synchronization manifold, analytical eigenvalue placement has proved unsuccessful for now. Instead, Lyapunovs direct method and LaSalles’ invariance principle have been used to investigate the stability of the synchronization manifold. The results are summarized in Theorem 8.1.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

8.2 Rotating Discs without Eccentric Masses

45

Theorem 8.1 (Global Asymptotic Stability of the Synchronization Manifold: Rotating Discs (no eccentric masses)). Consider the system of nonlinear differential equations (8.4) - (8.6) with νi = 0 i = 1, 2, 3 and define the manifold S ⊆ S × S × S × R × R × R as the set S = {[θ θ ′ ]T ∈ S × S × S × R × R × R|θ1 = −θ2 , θ1′ = −θ2′ , θ3′ = θ3 = 0}. Assume, furthermore, identical oscillator properties and no eccentric masses: 1. ̟1 = ̟2 . 2. µ1 = µ2 . 3. ν1 = ν2 = 0. Then the system (8.4) - (8.6) will converge to S as t → ∞ for all initial conditions. Proof. The proof is provided in Appendix A.3. 8.2.1

Simulation and Experimental Results

Figure 8.2 shows an example of simulation results yielding the synchronizing dynamics of (8.4) (8.6) with νi = 0 i = 1, 2, 3 and using the parameter set provided in Table 8.2. As becomes clear from Figure 8.2 the oscillators are locked in anti-phase synchronized motion after τ ≈ 450 [−] and the oscillators move with equal amplitudes, while the beam comes to a complete standstill. Table 8.2: Simulation and experimental parameters, rotating discs (without eccentric masses). Dimensionless Non Dimensionless ω1 = ω2 = 7.0711 [rad s−1 ] ̟1 = ω2 = 1 ω3 = 3.1623 [rad s−1 ] ̟3 = 0.4472 ωe,i = 0, i = 1, 2, 3 νe,i = 0, i = 1, 2, 3 µ1 = µ2 = 0.1000 µ1 = µ2 = 0.1000 [−] ζ3 = 0.0707 [−] ζ3 = 0.0707

0.02

0

−0.02

−0.04

0

50

100

150

200

250

300

350

400

450

0.01

ξ3 [−]

0.005

∆φ1,2 (×π) [rad]

ξ1 , ξ2 [−]

0.04

1.2

1

0.8

0.6

0.4

0

0.2 −0.005

−0.01

0

50

100

150

200

250

τ [−] (a)

300

350

400

450

0

0

50

100

150

200

250

300

350

400

450

τ [−] (b)

Figure 8.2: Synchronization between two rotating discs (simulation results) θ0 = ′ ′ ′ T [θ10 θ10 θ20 θ20 θ30 θ30 ] = [−0.05 0 − 0.04 0 0 0]T (a) Transient behavior (Top: left and right disc, Bottom: center disc) (b) Phase difference between the left and right disc.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

46

8.2 Rotating Discs without Eccentric Masses

Next, an experiment is conducted with the same parameter set and initial conditions. The results are depicted in Figure 8.3. Again synchronizing dynamics are observed, according to Definition 5.3 (α = 1.3333, ε = 3.5 · 10−3 ). However, the steady state amplitudes of the oscillators are no longer equal and the steady state phase difference is not exactly equal to π nor does it converge to one value. Instead it oscillates around approximately 0.95π. In the next paragraph, these observations are discussed in more detail. 1.2

0.05

0

−0.05

−0.1

0

50

100

150

200

250

300

350

400

0.02

∆φ1,2 (×π) [rad]

ξ1 , ξ2 [−]

0.1

1

0.8

0.6

0.4

ξ3 [−]

0.01

0

0.2 −0.01

−0.02

0

50

100

150

200

250

300

350

400

0

0

50

τ [−] (a)

100

150

200

250

300

350

400

τ [−] (b)

Figure 8.3: Synchronization between two rotating discs (experimental results) θ0 = ′ ′ ′ T [θ10 θ10 θ20 θ20 θ30 θ30 ] = [−0.05 0 − 0.04 0 0 0]T (a) Transient behavior (Top: left and right disc, Bottom: center disc) (b) Phase difference between the left and right disc.

Comparing the simulation results in Figure 8.2 and the experimental results in Figure 8.3 the following is observed. First, both numerical and experimental results show the same qualitative steady state behaviour. However, as in the case of coupled Duffing oscillators the (approximate) anti-phase synchronized state is reached faster in experiments (τ ≈ 200 [−]) than in simulations. A possible explanation is again the fact that disturbances might push the trajectory of the system away from the initial (approximately in-phase) mode in the experiments. Furthermore, compensating for the difference in dynamics of the oscillators in the set-up and finding the appropriate damping compensation proves rather challenging. Correspondingly, the initial increase in amplitude that is observed in Figure 8.3 shows that a nett energy input to the oscillators is still present. The fact that the oscillator amplitudes are not identical (due to non-identical oscillators) and the necessity to dissipate the energy overflow through the beam leads to a non-zero steady state motion of the beam. Finally, the steady state phase difference oscillates around approximately 0.95π. This is most likely caused by the combined influence of the non-zero motion of the beam and the nonidentical dynamics of the oscillators. Another possibility is that the (small) mismatches between the feedback linearizing controller LR (ξ, ξ ′ ) and the actual dynamics f (ξ, ξ ′ ) cause interference in the closed loop dynamics and thus influence the virtual dynamics. Such mismatches are likely to be different for each oscillator and might therefore add to the observed results. 8.2.2

Convergence Rate (Analysis and Simulation)

In this paragraph the synchronizing behaviour observed in both the experiments and simulations is explored in more detail. Attention is focussed on the synchronization time (defined in more detail later). Since the system (8.4) - (8.6) with νi = 0 i = 1, 2, 3 is linear, dependence of synchronization time can be explored by examination of the eigenvalues of the system after a suitable coordinate transformation.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

8.2 Rotating Discs without Eccentric Masses

47

Consider the system (8.4) - (8.6) and define the sum variable δ = θ1 + θ2 . The system is written in a new set of state space coordinates δ = [δ δ ′ θ3 θ3′ ]T : δ ′ = A δ, where



0  −̟2 A=  0 µ̟2

 1 0 0  0 2̟2 0 .  0 1 0 0 −(2µ̟2 + ̟32 ) −2ζ3 ω3

(8.10)

(8.11)

In theory even a perfect system (identical oscillators and no disturbances) does not reach a complete synchronized state in finite time, when started from an initial condition that is not exactly in antiphase. In order to be able to define such a thing as synchronization time it is therefore necessary to define a measure of sufficient synchronization. This also allows the study of systems that do not synchronize completely, such as any experimental system. Definition 8.1 (Synchronization time). The synchronization time τs of a system is defined such that for any τ > τs : δ(τ ) < ε. (8.12) In the sequel ε = 10−4 , since simulations show that the phase difference between the oscillators has converged to sufficiently towards π if δ(τ ) < 10−4 . From Theorem 8.1 it is known that the anti-phase synchronization manifold is globally asymptotically stable. I.e. in system (8.10) the origin is globally asymptotically stable, hence A is Hurwitz for positive parameter value. A direct proof that A is Hurwitz has not been acquired yet, since attempts to directly calculate the eigenvalues failed and more conservative conditions like Gersgorins’ circle theorem cannot be used either. The convergence rate of the system is expected to be inversely proportional to the real part of the slowest eigenvalue λ− δ . This is the eigenvalue of A with the smallest real part in absolute sense (closest to zero). In this report only a single study of parameter dependence of the convergence rate is presented by means of simulation. More extensive studies and experimental verification is left to future studies. In the following, the dependence of the synchronization time τs on the damping of the beam ζ3 is investigated. In order to study this dependence, the parameter set provided in Table 8.3 has been selected. Using these parameters the eigenvalues of the system matrix A are calculated for ζ3 ∈ (0 2.5]. Now define the inverse of the real part, of the slowest eigenvalue as: 
−1 υ = |ℜ λ− | , δ

(8.13)

Solving the system (8.4) - (8.6) numerically for the same parameter set and a range of ζ3 allows the calculation of τs according to Definition 8.1. Scaling both the simulation and the eigenvalue analysis results between 0 and 1 yields Figure 8.4. Table 8.3: Parameters for simulations and eigenvalue analysis concerning the dependence of synchronization time on the dimensionless damping of the beam. Dimensionless Non Dimensionless ω1 = ω2 = 6.2832 [rad s−1 ] ̟1 = ω2 = 1 ̟3 = 0.7252 ω3 = 4.5564 [rad s−1 ] ωe,i = 0, i = 1, 2, 3 νe,i = 0, i = 1, 2, 3 µ1 = µ2 = 0.0256 [−] µ1 = µ2 = 0.0256

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

48

8.3 Rotating Discs with Eccentric Masses

0

10

τs⋆ [−]

υ Simulation

−1

10

0

0.5

1

1.5

2

ζ3 [−] Figure 8.4: Comparison between analytical and simulation results with respect to the dependence of scaled synchronization time τs⋆ on the dimensionless damping of the beam ζ3 . The behaviour observed from Figure 8.4 may be understood as follows. For the theoretical situation that ζ3 = 0 no synchronization will occur since every initial condition results in an unique trajectory and the system will stay on that torus in state space for all time and any initial mode is stable. As ζ3 increases the system will synchronize faster since the energy dissipation is the driving force behind synchronization. However, for large values ζ3 , synchronization will tend to take longer again since the interaction between the oscillators starts to be limited by the heavily damped motion of the beam. Combining the reasoning for small and large ζ3 it becomes clear that there is an optimal value of ζ3 for which synchronization occurs the fastest. For the given parameter set both the analysis and the simulation show that this value is given by ζ3 ≈ 0.39. The difference between the simulations and the eigenvalue analysis is most probably caused by difference between the numerical definition of approximate synchronization versus the semi-analytical calculation of the systems’ slowest timescale.

8.3

Rotating Discs with Eccentric Masses

To conclude the discussion of the system (8.4) - (8.6) this paragraph discusses the dynamics of the complete system, i.e. νi 6= 0. Starting with a preliminary analysis of the stability of the synchronization manifold, attention is focussed on numerical and experimental results. The chapter is then concluded with a short summary and some remarks. The analytical results obtained for this system do not allow a rigorous proof of the existence and stability of synchronization regimes. Although this type of analysis proved successful for different types of systems and the case where νi = 0 it proved unsuccessful to apply LaSalles invariance principle for the case where νi 6= 0 till now. These results are summarized in conjecture 8.1. Conjecture 8.1 (Global Asymptotic Stability of the Synchronization Manifold: Rotating Discs (with eccentric masses)). Consider the system of nonlinear differential equations (8.4) - (8.6) and define the manifold S ⊆ S×S×S×R×R×R as the set S = {[θ θ ′ ]T ∈ S×S×S×R×R×R|θ1 = −θ2 , θ1′ = −θ2′ , θ3′ = θ3 = 0}. Assume, furthermore, identical oscillator properties: 1. ̟1 = ̟2 . Huygens Synchronization in Various Dynamical Systems. Experimental Results.

8.3 Rotating Discs with Eccentric Masses

49

2. ν1 = ν2 . 3. µ1 = µ2 . Then the system (8.4) - (8.6) will converge to S as t → ∞ for all initial conditions. Argumenation. Consider the system (8.1) - (8.3). To analyze the limit behaviour of this system, consider the total energy as a candidate Lyapunov function: ∆θi 3 Z 3 3 X 1 X ˙2 X ki s ds me gℓ(1 − cos(θi ) + V= Ji θi + 2 i=1 i=1 i=1

(8.14)

0

Calculating the derivative of V along the solutions of the system (A.27) - (A.29) yields: V˙ =

3 X i=1

Ji θ¨i2 θ˙i +

3 X

me gℓ sin(θi ) +

i=1

3 X i=1

ki ∆θ˙i ∆θi = −2ζ3 ω3 θ˙32 .

(8.15)

Hence, V˙ ≤ 0 and the system may be analyzed using LaSalles invariance principle. This far, the reasoning is identical to the proof in Appendix A.3 for the case where νi = 0, i = 1, 2, 3. However, in the case where νi 6= 0 the arguments used guarantee anti-phase synchronized motion on the subset where V˙ = 0, fail. The conjecture is however, supported by both experimental and numerical results. Furthermore, it is based on previous experience with similar dynamical systems where it was possible to provide rigorous proof of global asymptotic stability of the anti-phase synchronization manifold. Especially the proof in Appendix A.3 for the case where νi = 0 supports this conjecture. Although no rigorous proof has been found yet, conjecture 8.1 suggest that anti-phase synchronization is globally attractive for the system (8.4) - (8.6). In order to make the conjecture more plausible the final part of this chapter supplies both numerical and experimental results that indeed suggest anti-phase synchronized motion to be a stable and attractive mode of the system. 8.3.1

Simulation and Experimental Results

In order to show that the behaviour of (8.4) - (8.6) does converge to anti-phase synchronized motion both numerical and experimental results are provided. It is not the objective to proof conjecture 8.1 this way. However, the results add to the validity of the conjecture. The experiments and simulations are conducted using the parameters provided in Table 8.4. Figure 8.5 shows the simulation results, while Figure 8.6 shows the corresponding experimental results. Table 8.4: Simulation and experimental parameters, rotating discs with eccentric masses. Non Dimensionless Dimensionless ω1 = ω2 = 6.6667 [rad s−1 ] ̟1 = ω 2 = 1 ̟3 = 0.4568 ω3 = 3.0455 [rad s−1 ] ωe,1 = ωe.,2 = 4.6690 [rad s−1 ] νe,1 = νe,2 = 0.7004 ωe,3 = 2.3847 [rad s−1 ] νe,3 = 0.3577 µ1 = µ2 = 0.1043 µ1 = µ2 = 0.1043 [−] ζ3 = 0.1523 [−] ζ3 = 0.1523

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

50

8.3 Rotating Discs with Eccentric Masses

0.02

0

−0.02

−0.04

0

100

200

300

400

500

600

700

800

900

0.01

∆φ1,2 (×π) [rad]

ξ1 , ξ2 [−]

0.04

ξ3 [−]

0.005

1.2

1

0.8

0.6

0.4

0

0.2 −0.005

−0.01

0

100

200

300

400

500

600

700

800

0

900

0

100

200

300

400

500

600

700

800

900

τ [−]

τ [−] (a)

(b)

Figure 8.5: Synchronization between two rotating (eccentric) discs (simulation results) θ0 = ′ ′ ′ T [θ10 θ10 θ20 θ20 θ30 θ30 ] = [−0.05 0 − 0.04 0 0 0]T (a) Transient behavior (Top: left and right disc, Bottom: center disc) (b) Phase difference between the left and right disc.

0.05

0

−0.05

−0.1

0

50

100

150

200

250

300

350

400

0.02

∆φ1,2 (×π) [rad]

ξ1 , ξ2 [−]

0.1

1

0.8

0.6

0.4

ξ3 [−]

0.01

0

0.2 −0.01

−0.02

0

50

100

150

200

τ [−] (a)

250

300

350

400

0

0

50

100

150

200

250

300

350

400

τ [−] (b)

Figure 8.6: Synchronization between two rotating (eccentric) discs (experimental results) θ0 = ′ ′ ′ T [θ10 θ10 θ20 θ20 θ30 θ30 ] = [−0.05 0 − 0.04 0 0 0]T (a) Transient behavior (Top: left and right disc, Bottom: center disc) (b) Phase difference between the left and right disc.

When analyzing the data provided 8.5 and 8.6 it becomes clear that although the simulations respond as expected the experiments deviate from the response that one might expect initially. Note that both the simulation and experiment converge to a phase locked mode as time evolves. The final phase difference in the simulations is close to anti-phase, i.e. ∆φ1,2 ≈ π. However, the experiment converges to a state where ∆φ1,2 ≈ 0.8π (approximate anti-phase synchronization according to Definition 5.3 with α = 1.4068 and ε = 5.9·10−3 ). The explanation for this discrepancy may (as in previous results) be found in the fact that during the experiment it proved very hard to find the correct energy input for the system. The trade-off between a positive energy input to account for the non identical oscillators and to compensate for damping on the one hand, and the

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

8.4 Concluding Remarks

51

fact that any nett energy input will result in a non-zero steady state motion of the beam on the other hand, does not allow the experimental system to approach the anti-phase synchronization manifold as close as the simulations. The reason that the difference is larger in this case than in previous experiments is not clear. However, tuning the energy input for the system even finer might yield better results. The difference in amplitude between the oscillators is again caused by the fact that the oscillators are not identical. Finally, in the simulations approximate synchronization is reached after τ ≈ 900 [−] while the experiments show a much faster convergence and reach the phase locked state after τ ≈ 60 [−]. As in the previous examples this difference might be explained by the fact that external disturbances push the system away from the approximate in-phase initial conditions in the experiments, while these disturbances are not present in the simulations.

8.4

Concluding Remarks

In the this chapter a system of coupled rotating discs has been analyzed. First, a simplified version of the dynamics has been considered in the absence of eccentric masses. This allowed for the analysis of the stability of the synchronization manifold. Next to the analytical results, both numerical and experimental results are provided. Moreover, the simplified system has been used to demonstrate a preliminary study of the parameter dependence of the convergence rate of the system. Finally, the complete system has been considered and a conjecture about the stability of the synchronization manifold has been put forward. Supporting this conjecture, numerical as well as experimental results have been presented.

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52

9

9 Huygens Clocks

Huygens Clocks

This final chapter of the report discusses preliminary results with respect to modeling the classical Huygens set-up. Starting with the introduction of the dynamical system and a theoretical study of the stability of the synchronization manifold, the chapter continues with simulation results. Unlike the preceding chapters no experimental results are provided since it has not been possible to successfully conduct experiments within the time frame of this project. Instead a discussion of the observed problems and their most probable causes is provided. Naturally, this leads to a set of suggestions for future study in order to allow successful modeling of the Huygens set-up.

9.1

Equations of Motion and Controller Design x k g

M b ℓ1 θ1

ey

ℓ2 θ2

e1n

e2n

e1t ex

e2t m2

m1

ez

Figure 9.1: Schematic representation of Huygens set-up.

Figure 9.1 depicts the classical Huygens set-up as discussed from a historical perspective in Chapter 2. The systems consist of two pendula mounted on a common frame. The beam has one degree of freedom in horizontal direction and its motion is constrained by a linear spring and damper. The equations of motion of this system are: θ¨1

=

θ¨2

=

x ¨ =

where ωi =

q

g ℓi

x¨ cos(θ1 ) − ω12 sin(θ1 ) ℓ1 x¨ − cos(θ2 ) − ω22 sin(θ2 ) ℓ2 2 i X µi h ¨ − ˙ ℓi θi cos(θi ) − θ˙i2 sin(θi ) − ω32 x − 2ζω3 x, 3 i=1

(9.1)

−

i = 1, 2 the eigenfrequency of the pendula, ω3 = b3 2(m1 +m2 +M)ω3

q

k m1 +m2 +M

(9.2) (9.3)

the eigenfrequency

of the beam/pendula combination, ζ = the dimensionless damping of the beam and mi µi = 3 m1 +m2 +M the mass ratio between the beam and the pendula, scaled such that µ ∈ [0 1], as in the previously discussed case studies. In order to write the system of equations (9.1) - (9.3) in dimensionless form, the following set of dimensionless parameters is defined:

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

9.1 Equations of Motion and Controller Design

53

Table 9.1: Dimensionless parameters, Huygens clocks. τ∗ = ω ¯t ωi = ω ¯ ̟i∗ , i = 1, 2 ω3 = ω ¯ Ω∗ x = ℓξ = ω¯g2 ξ ∗ Here, ω ¯ = 21 (ω1 + ω2 ) and ∗ indicates a dimensionless parameter which will be omitted in the sequel for the sake of readability. The resulting dimensionless equations of motion are: θ1′′ θ2′′

= =

−̟12 (ξ ′′ cos(θ1 ) + sin(θ1 )) −̟22 (ξ ′′ cos(θ2 ) + sin(θ2 ))

ξ ′′

=

−

2 X µi i=1

3

  ̟−2 θi′′ cos(θi ) − θi′ 2 sin(θi ) − Ω2 ξ − 2ζΩξ ′ ,

(9.4) (9.5) (9.6)

An analysis of the stability of possible synchronization regimes in the system (9.4) - (9.6) is provided in (Pogromsky et al., 2003). In order to make this report self contained these results are repeated in Theorem 9.1. Theorem 9.1 (Global Asymptotic Stability of the Synchronization Manifold: Huygens Clocks). Consider the system of nonlinear differential equations (9.4) - (9.6) and define the manifold S ⊆ S × R × S × R × R × R as the set S = {[θ1 θ1′ θ2 θ2′ ξ ξ ′ ]T ∈ S × R × S × R × R × R|θ1 = −θ2 , θ1′ = −θ2′ , ξ ′ = ξ3 = 0}. Assume, furthermore, identical oscillator properties: 1. ̟1 = ̟2 . 2. µ1 = µ2 . Then the system (9.4) - (9.6) will converge to S as t → ∞ for all initial conditions. Proof. The proof is provided in (Pogromsky et al., 2003) and is obtained along the same lines as the proofs provided in Appendix A.2 and A.3. Now that the dynamics of the Huygens system have been introduced and the existence of a globally asymptotically stable synchronization manifold has been shown, simulation results are provided that illustrate the synchronizing dynamics of the system. The simulations are conducted using the parameter set provided in Table 9.2. Figure 9.2a shows the response of the system, while Figure 9.2b shows the evolution of the phase difference between the oscillators. Table 9.2: Simulation parameters, Huygens clocks. Dimensionless Non Dimensionless ω1 = ω2 = 3.1321 [rad s−1 ] ̟1 = ̟2 = 1 Ω = 1.3034 ω3 = 4.0825 [rad s−1 ] ζ3 = 0.2041 ζ3 = 0.2041 µ1 = µ2 = 0.2500 µ1 = µ2 = 0.2500

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

54

9.2 Problems and Solutions Concerning Experimental Results

θi [rad]

0

−0.5

0

50

100

150

200

250

300

350

400

0.15

∆φ1,2 (×π) [rad]

1.2 0.5

1

0.8

0.6

0.1

ξi [−]

0.4 0.05 0

0.2

−0.05 −0.1 0

50

100

150

200

250

300

350

400

0

0

50

100

150

200

250

300

350

400

τ [−]

τ [−] (a)

(b)

Figure 9.2: Synchronization between two pendula in the Huygens set-up (simulation results)
[θ1 θ1′ θ2 θ2′ θ3 ξ ξ ′ ]T = [− π4 0 − π4 + ǫ 0 0 0]T with ǫ = 0.1 · π4 (a) Transient behavior (Top: red and blue oscillator, Bottom: beam) (b) Phase difference between the oscillators. As predicted by Theorem 9.1 the dynamics of (9.4) - (9.6) converge to a state of anti-phase synchronized motion and reach a state of approximate anti-phase synchronization after τ ≈ 350 [−]. In this mode the beam comes to a complete standstill and the oscillators move with equal amplitude and frequency.

9.2

Problems and Solutions Concerning Experimental Results

This paragraph discusses the current state of events of this part of the project. Although, it proved unsuccessful to conduct synchronization experiments with the modification to the set-up, the developed controller is derived for future reference. The chapter concludes with discussion of the current understanding of the problems observed and suggestions for possible causes and solutions. Controller Design In order to realize a virtual dynamical system that mimics the dynamics prescribed by (9.4) - (9.6) the masking dynamics approach discussed in Chapter 6 is applied. The following identities are proposed to realize the required dynamics: LH (ξ, ξ ′ ) = ′

DH (η, η ) = GH =

−f (ξ, ξ ′ , v = 0) Λ−1 F(η, η ′ ) Λ−1 ,

(9.7) (9.8) (9.9)

where Λ = diag[λ1 λ2 λ3 ] contains the gains from (4.4) - (4.6). The block ξ ′′ = LH (ξ, ξ ′ ) = −f (ξ, ξ ′ , v = 0) Λ−1 cancels the set-ups’ dynamics ξ ′′ = f (ξ, ξ ′ , v). The block η ′′ = DH (η, η ′ ) = FH (η, η ′ ), furthermore, assures that the closed loop dynamics of the complete system equal those of the user specified open loop dynamics η ′′ = FH (η, η ′ ) which are specified by (9.4) - (9.6). The virtual coordinate system η = [θ1 θ2 ξ]T is obtained from choosing the appropriate coordinate transformation TH = diag[θm θm 1], where θm is the angle of rotation corresponding to the stroke of the set-up. Note that the dynamics η ′′ = DH (η, η ′ ) = FH (η, η ′ ) are specified by (9.4) - (9.6) which is are directly and indirectly dependent on the states [θ1 θ1′ θ2 θ2′ ξ ξ ′ ]T of the system. The equations depend directly on the accelerations of both the beam and the oscillators. Determining these Huygens Synchronization in Various Dynamical Systems. Experimental Results.

9.2 Problems and Solutions Concerning Experimental Results

55

accelerations numerically during the experiment is highly sensitive to noise and results in barely usable results. Instead it is possible to derive a direct relation between the states and the acceleration of the beam ξ ′′ from (9.4) - (9.6). After substitution and some trigiometry this yields:

ξ ′′ (θi , θi′ , ξ, ξ ′ ) =

µ 3

2  P 1

i=1

2

 sin(2θi ) + ̟i−2 θi′ 2 sin(θi ) − 2ζΩξ ′ − Ω2 ξ 1−

µ 3

2 P

cos2 (θ

.

(9.10)

i)

i=1

Given (9.10) the acceleration of the pendula follows directly from (9.4) - (9.5). Since no successful synchronization experiments have been conducted yet, the chapter is concluded by a discussion of the observed problems and suggestions for improvements in future research.

Discussion Problem Analysis At this point the derived controller dynamics have successfully been implemented in Simulink and it is possible to conduct experiments like the ones discussed in the preceding chapters using the virtual system. The results are however, not as expected. For those parameter sets where it was possible to obtain regular oscillating motion, the oscillators appear to move completely independent of each other. In short, the controller seems to work but no synchronization is observed, even if the system is started very close to the anti-phase synchronization manifold. Based on the results, a number of possible explanations may be put forward. A very straightforward option is that the controllers are not correct, or not correctly implemented. Given the successful experience with the Duffing oscillators and the rotating discs this does however, not seem very plausible. Another possibility is that no suitable parameter set has been selected during the trials. However, both analytical results and simulations show that synchronization is a very robust phenomenon which makes this explanation rather unlikely as well. The most likely causes for the problems with modeling the Huygens set-up, using the masking approach are the following: First of all, in the Huygens set-up the oscillators and the beam are only coupled by acceleration forces. Any deviation of the feedback linearizing controller with respect to the actual dynamics of the set-up therefore causes a non-physical coupling between pendula rotation and beam translation that is not possible in the Huygens set-up. The combined influence of this coupling with that of the classical Huygens set-up which is added in the virtual dynamics is hard to predict. The final possible cause for the deviating experimental results is the fact that the present actuators only allow the modeling of small amplitude, slow oscillations, because their actuator power is limited. If the frequency or amplitude of the oscillator is increased the maximum velocity rises as well. Correspondingly, the required accelerations increase and therefore the required actuator power has to be increased. The present actuators only allow for experiments where the maximum oscillating frequency is about 1 − 3 [Hz] while the amplitude is in the order of magnitude of 2 [deg]. It is believed that the coupling forces caused by these small and slow oscillations is small compared to the disturbances and model uncertainties in the closed loop system. The fact that these slow and small amplitude oscillations lead to synchronization in the previous chapter can be understood by the fact that the system of rotating discs considered in that case does not suffer from the residual non-physical coupling resulting from model uncertainties in the feedback linearizing controller. Solutions As discussed in the preceding paragraph the present actuators in the set-up do not meet the requirements that are needed in the experiments. This chapter is concluded with a number of suggestions that will possibly allow for successful experiments in the future. First, note that the required actuator power follows directly from the experiments that need to be conducted. Therefore it is highly Huygens Synchronization in Various Dynamical Systems. Experimental Results.

56

9.3 Concluding Remarks

recommended to make a detailed overview of the experiments that are required and to investigated the required actuator power by simulation before investing in a new actuator system. Focussing on the Huygens set-up two alterations are expected to improve experimental results. First, increasing the oscillator frequency will increase the coupling, since it is directly related to the acceleration of the oscillators and thus the amount of interaction between the oscillators though the beam. Second, increasing the virtual rotation angles represented by the stroke of the set-up by choosing a different mapping also results in higher oscillator speeds. Both of these recommendations require higher accelerations of the masses in the set-up and therefore require stronger actuators. To get a feeling for this, consider the Huygens system with oscillator masses m1 = m2 = 50 [g] and a beam of mass M = 800 [g]. With k = 70 [N m−1 ], b = 10 [N sm−1 ] and ℓi = 0.25 [m]. Both simulations and experiments show that the required actuator power is well within the present limits in this case. Mapping the displacement of the oscillators directly to the arc traveled by the oscillator ends results in oscillations with an amplitude of approximately 1 [deg] with an oscillating frequency of approximately 1 [Hz]. If the coupling is increased by lowering ℓi (increasing the natural frequency of the oscillators) simulations show that in order to obtain an oscillation frequency of 5 [Hz] the required actuator power is about 275% of the power present in the system. If instead the pendula length is kept constant, but the the displacements of the oscillators in the set-up are magnified, thus leading to larger virtual rotations and a stronger interaction between the oscillators, the required actuator power to allow for the oscillations with an amplitude of approximately 45◦ is 250% of the current actuator power. The previous example illustrates the problems discussed in this chapter. Note that next to the actuator power required to ’drive’ the virtual dynamics a significant percentage of the actuator power is required to allow for compensation of the natural stiffness and damping of the set-up. Although the stiffness is not expected to cause any significant problems (it is bounded by the same maximum, no matter which virtual dynamics are required), the damping increases with the speed of the oscillators which causes an increase in actuator power as well, when considering faster or larger amplitude oscillations. Therefore, considering a new actuator system that does not suffer from electrical damping as severely as the present one serves two goals at once. First of all, the actuators may be chosen at the appropriate strength and secondly, less of the actuator power is required to cancel the system’s dynamics.

9.3

Concluding Remarks

In this chapter a number of problems that have been observed when modeling the Huygens set-up have been discussed. The most important conclusion is that the limited actuator power in the system causes severe limitations in experiments. Although the chapter aims to illustrate these problems it does not provide a clear definition of the required actuator system. The reason for this is that the required actuator power depends mainly on the required experiments and therefore should be selected only after these have been carefully selected. Next to stronger actuators it is important to consider actuator systems that suffer from electrical damping as little as possible since in doing so, one increases both the portion of actuator power that can be used to model the virtual dynamics and reduces the amount of interference that is caused by inexact damping compensation.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

10 Conclusions and Recommendations

10 10.1

57

Conclusions and Recommendations Conclusion

This thesis aims at: ’Investigating, using a new experimental set-up, the existence and stability of synchronization regimes in coupled oscillatory systems.’ It consists of two main parts. In the first part of the thesis, a new experimental set-up, designed at the departement of Mechanical Engineering of the Eindhoven University of Technology, is described. The actual development of the necessary hard- and software, vital to getting this set-up operational, is a significant part of the thesis research and is also described in this part. A model for the dynamics of the set-up is developed and discussed in detail. The final part of the thesis focuses on the experimental results obtained with the experimental set-up. These results are accompanied by analytical and numerical results, providing a compete analysis of the observed synchronization phenomena. The following paragraphs describe the results presented in this report in further detail. Hard- and Software Development The experimental set-up has been designed and constructed previous to the start of the project described in this thesis. The hard- and software needed to operate the set-up were, however, not yet available at that time. An important part of thesis research is, therefore, devoted to developing these essential components. The first step has been the selection en implementation of the data acquisition system. A safety system has, furthermore, been constructed, allowing the set-up to run without the need of human presence. Finally, a combination of Simulink models and Matlab code has been developed, facilitating the operation of the set-up. A fully automated calibration routine, for example, enables fast and easy initialization of the set up. The possibility to detect synchronization online also minimizes the efforts needed to successfully conduct experiments. Modeling of the Set-up This thesis describes a set of experiments specifically aiming to tune the actuator / amplifier strengths within the experimental set-up. An experimental and theoretical study of damping and hysteresis effects has also been carried out. Finally, a model for the systems dynamics has been developed and described. Both the qualitative and quantitative properties of this model closely match those of the experiments conducted specifically for this thesis. A detailed description and a comparison with experimental results obtained using the experimental set-up, are given in the main text of this thesis. An overview of possible nonlinear effects in the set-up is presented as well. Development of Virtual Closed Loop Dynamics A controller, based on a feedback linearization procedure and nonlinear state feedback, has been developed in the course of the thesis. The purpose of this is to enable experiments with dynamics other than those inherit to the experimental set-up. This feedback ’neutralizes’ the dynamics of the set-up and ’replaces’ it with a set of user-specified dynamics. The result is a virtual system, enabling experiments with a wide range of dynamical systems, using only one experimental set-up as a platform. A number of potential drawbacks of this approach, i.e. sensitivity to distortions and uncertainties in the model parameters, are also discussed. Experimental Results In the final part of this thesis, the experimental results of synchronizing dynamics in coupled dynamical systems are presented. Starting with the experimental results using the uncontrolled set-up, the thesis continues with a description of the results using a range of dynamical systems. A system of coupled Duffing oscillators is described and analytical results are presented that show the synchronization manifold to be globally asymptotically stable. Numerical and experimental results are also presented, supporting these results. Next, two sets of experiments are described, illustrating the Huygens Synchronization in Various Dynamical Systems. Experimental Results.

58

10.2 Recommendations

existence of synchronization regimes in a dynamical system of coupled rotating discs. For this last system, however, stability of the synchronization manifold has not been proven yet. A conjecture, is formulated, predicting the anti-phase synchronization manifold to be globally asymptotically stable. For a simplified version of the rotating discs system, however, a rigorous proof of stability of the synchronization manifold has been found. An analysis of the convergence rate of this system is given as well. This analysis indicates that the contraction rate of the system can be maximized in a nontrivial way by altering the damping in the system. Finally, an attempt is made to model the classical Huygens set-up. It has thus far not been possible, unfortunately, to experimentally show synchronization in this specific situation. A detailed overview of the encountered problems, possible reasons and possible solutions is presented in the main text. The next section will discuss these possibilities in more detail, while focusing on the recommendations for future research. Huygens Synchronization The research presented in this thesis shows that Huygens or frequency synchronization arises in a variety of dynamical systems. These results not only yield Huygens synchronization to occurs, but also shows that this phenomenon is very robust. Using simulations and experimental results as well as analytical results it has been shown that this type of synchronization is robust with respect to disturbances and non-identical oscillator properties. These results agree with studies presented in literature and with the original findings of Christiaan Huygens himself.

10.2

Recommendations

The experiments described in this thesis are a first step in the research of synchronization phenomena making use of this specific experimental set-up. The project yields important insights into both the potential of the experimental set-up and into the (possible) weaknesses / problems that can be encountered while working with such a set-up. This chapter concludes the thesis with a set of recommendations for future experiments, particulary listing a number of suggestions for improving the performance of the set-up. Future experiments can also combine the advantages of a fully actuated system with the derived control strategy. This facilitates the modeling of different dynamics, yielding a wide range of potential experiments. Future studies could include a detailed analysis of parameter influences on synchronization. The coupling strength, for example, could be altered both electrically and mechanically, leading to very interesting results. It is expected that this parameter in particular influences the type of behaviour found in experiments and to influence the convergence rate as well. For which minimal coupling strength, for example, will synchronization start to occur in practice? The influence of initial conditions of the experiment or different coupling laws could also be investigated, using the method and experimental set-up described in this thesis. Finally, the set-up is very suitable for experiments with controlled synchronization, opening up a completely new potential field of research. Modeling and Analysis The model of the experimental set-up described in this thesis provides a good estimate of the dynamics of this set-up. The compensation for damping / friction, however, proved to be a problem throughout the experiments. The system is very sensitive to the linear velocity feed-forward mechanism presently used. In order to obtain better and more consistent experimental results, an alternative escapement mechanism should be developed. If the present actuator system were replaced, as suggested later on, part of this problem would already be solved, since a significant part of the damping originates in the actuators. Furthermore, a rigorous proof of Conjecture 8.1 is believed to exist, based on the results presented in this thesis. A stability analysis proving this conjecture should be part of future studies. Finally, the preliminary results concerning the convergence rate of the system require experimental verification.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

59

Hardware Removal of S-shaped Springs In the course of this research, the hardware in the experimental set-up caused a number of problems. The S-shaped springs installed to facilitate the mechanical tuning of the eigenfrequencies of the oscillators and the beam, proved to be less useful than anticipated. The nonlinearity caused by these springs increases the complexity of the model of the set-up and has no known significant advantages. Moreover, the presence of these springs in the oscillators makes the oscillators less identical, leading to less precise experimental results. One of the primary goals of the thesis research is to facilitate experiments with different types of dynamical systems by means of a feedback controller. The underlying dynamics should therefore be as simple as possible. The required nonlinearities can, however, be added later through actuation. The S-shaped springs have already been removed from the oscillators. In future experiments, it is advisable to remove the S-shaped spring from the beam as well. This will further reduce the nonlinearity present in the system. It will, moreover, reduce the required actuator power to compensate for the stiffness in the beam. Removal of Dry Friction Elements The red oscillator in the experimental set-up shows stickslip behaviour. This is almost certainly caused by contact between the actuator coil and its casing. Another, less likely, possibility is that the contact takes place in the corresponding Linear Variable Differential Transformer (LVDT). Replacing the actuator system, as suggested later in this paragraph, will solve the problem of stick-slip behaviour. If this cannot be realized, the guidance of the red oscillator should be calibrated to make sure that there is no contact between the moving actuator elements throughout the oscillator stroke. Replacement of the Actuator System The present actuator system has two major drawbacks. Firstly, most of the damping observed in the system originates from back emf in the voice coils. The compensation for this damping poses a significant problem. These problems arise from the fact that actuator power is ’wasted’ on this compensation. Instead, the actuator power could be used more effectively while modulating the systems’ dynamics or applying other types of control. More importantly, damping should be minimized, since it is hard to identify and therefore contributes significantly to the model uncertainty. The present actuators have aluminum casings. These casings are the primary source of the damping in the system. Even when the coils are not connected, significant damping takes place. A system with polymer covers should, therefore, significantly improve the systems response. Secondly, the actuator power in the system should be increased. The range of experiments was significantly limited by the limited power of the present actuator system. At this moment, it is not possible to give a specific recommendation as to which actuator system should be used. The reason for this is that the choice of a new actuator system should be tailored to the experiments to be carried out. There is no doubt however, about a new actuator system needing to be significantly stronger than the present one. To conclude: A new actuator system not suffering from electrical damping as severely as the present one would serve two important purposes. First and foremost, the actuator could then be tuned to the appropriate strength such that a wider range of experiments could be conducted. Secondly, less of the actuator power would be required to cancel the system’s dynamics, since there would be less damping needing to be compensated for, leading to a more effective use of actuator power.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

60

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

61

Appendix

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

62

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

A Proofs and Derivations

A

63

Proofs and Derivations

A.1

Equations of Motion (Lagrange)

x3

x2

x1

k3

k1

b3

b1

u3

u1

k2 b2

m1

m2

u2 m3

Figure A.1: Schematic representation of the set-up.

Consider the system depicted in Figure A.1, where qi = qi (t) is the (relative) displacement of mass mi , i = 1, 2, 3, ki is a linear spring constant and bi a viscous damping constant. Linearity is assumed as a first approximation, in order to simplify the initial analysis. Furthermore ui = ui (t) represents the input supplied by the actuators. Note that two sets of coordinates will be used. First of all te set of relative coordinates q = [q1 q2 q3 ]T , since using these coordinates simplifies the form in which the equations of motion appear and these coordinates correspond to the displacements measured by the sensors in the set-up. Secondly, a set of absolute coordinates x = [x1 x2 x3 ]T will be used to represent the displacements and velocities with respect to a fixed reference frame, i.e. these are the motions that are observed when observing the set-up ’from the side line’. The set of absolute coordinates x is obtained from the relative coordinates q through a linear coordinate transformation T : R 7→ R, which in cartesian coordinates yeilds:   1 0 1 T : x = Tq , T = 0 1 1  (A.1) 0 0 1

Using a Lagrangian approach (Kraker and Campen, 2001) the equations of motion for this dynamical system will be derived. Define the following set of generalized coordinates q = [q1 q2 q3 ]T = ˙ is then given by: [x1 − x3 x2 − x3 x3 ]T . The total kinetic energy T (q)   m 0 m1 1 T 1  q˙ = 1 q˙ T M q, ˙ 0 m2 m2 ˙ = q˙ (A.2) T (q) 2 2 m1 m2 m1 + m2 + m3

with M is the mass matrix of the system. The potential energy stored in the springs is given by:   k1 0 0 1 1 V (q) = q T  0 k2 0  q = q T Kq, (A.3) 2 2 0 0 k3

with K the stiffness matrix of the system. Finally, the column of generalized non-conservative forces is derived using the principle of virtual work. • δq1 = [δq1 0 0]T

δW1 = (u1 (t) − b1 q˙1 ) δq1 = Qnc 1 δq1

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

64

A.1

• δq2 = [0 δq2 0]T • δq1 = [0 0 δq3 ]T

Equations of Motion (Lagrange)

δW2 = (u2 (t) − b2 q˙2 ) δq2 = Qnc 2 δq2 δW3 = (u3 (t) + u2 (t) − u1 (t) − b3 q˙3 ) δq3 = Qnc 3 δq3

Hence, the column of generalized non-conservative forces becomes:   u1 (t) − b1 q˙1  u2 (t) − b2 q˙2 Qnc =  u3 (t) − u2 (t) − u1 (t) − b3 q˙3

(A.4)

Furthermore, consider the following derivatives according to (Kraker and Campen, 2001): d (T,q˙ ) dt T,q V,q

= M q¨

(A.5)

= 0 = Kq.

(A.6) (A.7)

According to Lagrange the equations of motion now become: d (T,q˙ ) − T,q +V,q = (Qnc )T . dt

(A.8)

Hence, using previous results the equations of motion appear in a familiar form: M q¨ + B q˙ + Kq = F (t),

(A.9)

with M , K the mass and stiffness matrix as specified earlier, B the damping matrix and F (t) the column of actuation forces, according to:     b1 0 0 u1 (t)  u2 (t) B =  0 b2 0  F (t) =  (A.10) 0 0 b3 u3 (t) − u2 (t) − u1 (t) The system of ordinary differential equations (A.9) may be written in absolute coordinates x. This yields: x ¨1

=

x ¨2

=

x ¨3

=

−ω12 (x1 − x3 ) − 2ζ1 ω1 (x˙ 1 − x˙ 3 ) + c1 u1 (t)

−ω22 (x2 2 X i=1

where ωi =

q

ki mi

(A.11)

− x3 ) − 2ζ2 ω2 (x˙ 1 − x˙ 3 ) + c2 u2 (t)

(A.12)

  ˜3 (t), µi ωi2 (xi − x3 ) + 2ζi ωi (x˙ i − x˙ 3 ) − ω32 x3 − 2ζ3 ω3 x˙ 3 + c3 u

[rad s−1 ] the undamped eigenfrequency of the ith subsystem, ζi = th

bi 2mi ωi

mi m3 [−] the −2 −1

dimensionless damping of the i subsystem and µi = Furthermore, u ˜3 (t) = u3 (t) − u2 (t) − u1 (t) [V ] and ci [m s constants..

V

(A.13)

[−] the

dimensionless coupling strength. ] accounts for the amplifier / motor

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

A.2

A.2

Set-up: Stability of the Synchronization Manifold

65

Set-up: Stability of the Synchronization Manifold

Consider the following system of (nonlinear) coupled differential equations, representing the experimental set-up. x¨1 x¨2

= =

x¨3

=

−ω12 η1 (q1 ) q1 −ω22 η2 (q2 ) q2

2 X i=1

(A.14) (A.15)

  µi ωi2 ηi (qi ) qi − ω32 η3 (x3 ) x3 − 2ζ3 ω3 σ(q˙3 ) q˙3 ,

(A.16)

q ki mi ∈ R>0 , ζ3 ∈ R>0 , µi = m and mi ∈ R>0 ∀ i = where, qi = xi − x3 ∀ i = 1, 2, q3 = x3 , ωi = m i 3 1, 2, 3. Moreover, ηi (qi ), σ(q˙3 ) are (nonlinear) shape functions. Finally, let ηi (qi ) be an odd, one to one, continuous function, such that ηi (qi )qi has a zero only at qi = 0 and let q˙3 σ(q˙3 ) > 0 ∀ q˙3 6= 0 and σ(0) = 0. Note that the system used here is slightly different from the system in the main text. This is due to the fact that the proof is more comprehensible when using non-dimensionless variables. Furthermore, the constraints of undamped oscillators and the absence of actuator input has already been accounted for here and some notation is adapted to provide a more compact proof. However, the equations used in the proof are only scaled with respect to those in the main text. Therefore the results can directly be applied to the system of equations (5.4) - (5.6), which are in dimensionless form. In the following global asymptotic stability of the synchronization manifold is investigated using Lyapunovs direct method and LaSalles invariance principle. Theorem A.1 (Global Asymptotic Stability of the Synchronization Manifold (repeated in slightly modified form, from page 31)). Consider the system of nonlinear differential equations (A.14) - (A.16) and define S ⊆ R6 as the ˙ T ∈ R6 |x1 = −x2 , x˙ 1 = −x˙ 2 , x˙ 3 = x3 = 0}. Assume, furthermore, odd, one manifold S = {[x x] to one, continuous functions ηi (qi ), such that qi ηi (qi ) has a zero only at qi = 0 and σ(q˙3 ) such that q˙3 σ(q˙3 ) > 0 ∀ q˙3 6= 0 and σ(0) = 0. Finally, assume the following oscillator properties: 1. η1 (·) = η2 (·) 2. ηi (·) such that

Rx 0

3. ω1 = ω2

s ηi (s) ds → ∞ if |x| → ∞ i = 1, 2, 3

4. µ1 = µ2 . Then the system (A.14) - (A.16) will converge to S as t → ∞ for all initial conditions. Proof (Theorem 5.1: Global Asymptotic Stability of the Synchronization Manifold). Consider the system (A.14) - (A.16). To analyze the limit behavior of this system, consider the total energy as a candidate Lyapunov function: qi

3 Z 3 X 1X ki ηi (s) s ds mi x˙ 2i + V= 2 i=1 i=1

(A.17)

0

Calculating the derivative of V along the solutions of the system (A.14) - (A.16) yields: qZi (t) 3 X d mi x˙ i x¨i + V˙ = ki ηi (s) s ds. dt i=1 i=1 3 X

0

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

(A.18)

66

A.2

Set-up: Stability of the Synchronization Manifold

The terms in the second sum may be written as: d dt

Zqi

ki ηi (s) s ds = ki ηi (qi ) qi q˙i .

(A.19)

0

This yields: V˙ =

3 X

mi x˙ i x¨i +

3 X i=1

i=1

ki ηi (qi ) qi q˙i = −2ζ3 ω3 q˙32 σ(q˙3 ).

(A.20)

Hence, V˙ ≤ 0 and the system may be analyzed using LaSalles invariance principle. Equation (A.20) implies that V is a bounded function of time. Moreover, xi (t) is a bounded function of time and will converge to a limit set where V˙ = 0. On this limit set q˙3 = x˙ 3 = x¨3 = 0, according to (A.20) and thus q3 = x3 = x⋆3 is constant. Substituting this and (A.14) - (A.15) in (A.16) yields: x ¨1 + x ¨2 = −

x⋆3 ω3 η3 (x⋆3 ). µ

(A.21)

Integrating (A.21) twice with respect to time yields: x1 + x2 = −

x⋆3 ω3 η3 (x⋆3 )t2 + c1 t + c2 , µ

(A.22)

However, since both x1 (t) and x2 (t) are bounded functions of time, this yields x⋆3 = c1 = c2 = 0. Substituting x3 = x˙ 3 = x ¨3 = 0 in (A.16) shows: η1 (q1 ) q1 = −η2 (q2 ) q2

(A.23)

Since ηi (qi ) is an odd function and x3 = 0 on the limit set, this yields: x1 = −x2 ,

(A.24)

which, after differentiation with respect to time yields: x˙ 2 = −x˙ 1 .

(A.25)

Summarizing, it has been shown that as t → ∞ any solution will converge to the set where: x1 = −x2 ,

x˙ 1 = −x˙ 2 ,

x3 = x˙ 3 = 0

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

(A.26)

A.3

A.3

Rotating Discs: Stability of the Synchronization Manifold

67

Rotating Discs: Stability of the Synchronization Manifold

Consider the following system of nonlinear differential equations, representing the rotating disc system as discussed in Chapter 8: θ¨1 θ¨2

=

θ¨3

=

= −ω12 ∆θ1

i=1

with ∆θi = θi − θ3 , ωi = of inertia of the ith disc.

q

ki Ji ,

ζ3 =

(A.27)

−ω22 ∆θ2 2 X

(A.28)

  µi ωi2 ∆θi − ω32 θ3 − 2ζ3 ω3 θ˙3 ,

b3 2ω3 J3

Ji J3 ,

and µi =

(A.29)

where Ji = mi ℓi + Ji and Ji is the moment

In the following the stability of the synchronization manifold is investigated using Lyapunovs direct method and LaSalles invariance principle. As in the proof in Section A.2 the non-dimensionless form of the equations of motion is used since this makes the application of the total energy function of the system as a Lyapunov candidate more straightforward. Theorem A.2 (Global Asymptotic Stability of the Synchronization Manifold: Rotating Discs without Eccentric Masses (repeated in slightly modified from page 45)). Consider the system of nonlinear differential equations (A.27) - (A.29) and define the manifold ˙ T ∈ S × S × S × R × R × R|θ1 = −θ2 , θ′ = S ⊆ S × S × S × R × R × R as the set S = {[θ θ] 1 ′ ′ −θ2 , θ3 = θ3 = 0}. Assume, furthermore, identical oscillator properties: 1. ω1 = ω2 . 2. µ1 = µ2 . Then the system (A.27) - (A.29) will converge to S as t → ∞ for all initial conditions. Proof (Theorem 8.1: Global Asymptotic Stability of the Synchronization Manifold: Rotating Discs). Consider the system (A.27) - (A.29). To analyze the limit behavior of the system, consider the total energy as a candidate Lyapunov function: V=

3 3 1 X ˙2 X 1 Ji θi + ki ∆θi2 2 i=1 2 i=1

(A.30)

Calculating the derivative of V along the solutions of the system (A.27) - (A.29) yields: V˙ =

3 X i=1

Ji θ¨i2 θ˙i +

3 X

ki ∆θ˙i ∆θi ,

(A.31)

i=1

where the last term is found from direct evaluation of the integral and taking the derivative in (A.30). This yields: 2 (A.32) V˙ = −2ζ3 ω3 θ˙3 . ˙ Hence, V ≤ 0 and the system may be analyzed using LaSalles invariance principle. Equation (A.32) implies that V is a bounded function of time. Moreover, θi (t) is a bounded function of time and will converge to a limit set where V˙ = 0. On this limit set θ˙3 = θ¨3 = 0, according to (A.32) and thus θ3 = θ3⋆ is constant. Substituting this and (A.27) - (A.28) in (A.29), yields: ω3 ⋆ θ . µ 3

(A.33)

ω3 ⋆ 2 θ t + c1 t + c2 µ 3

(A.34)

θ1 + θ2 = − Integrating (A.33) twice with respect to time yields: θ1 + θ2 = −

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

68

A.3

Rotating Discs: Stability of the Synchronization Manifold

However, both θ1 (t) and θ2 (t) are bounded functions of time. This yields θ3⋆ = c1 = 0. Substituting θ3 = θ˙3 = θ¨3 = 0 in (A.29) shows: θ1 = −θ2 , (A.35) which, after differentiation with respect to time, yields: θ˙1 = −θ˙2 ,

(A.36)

Summarizing, it has been shown that as t → ∞ any solution will converge to the set where: θ1 = −θ2 ,

θ˙1 = −θ˙2 ,

θ3 = θ˙3 = 0

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

(A.37)

B Additional Data: System Identification

B B.1

69

Additional Data: System Identification Parameter Values Table B.1: Parameter values in equations (4.1) - (4.3). Oscillator 1 Oscillator 2 Frame / Beam (3 ) ωi [rad s−1 ] 12.5521 14.0337 9.7369 ζi [−] 0.3362 0.4296 0.0409 ci [m s−2 V −1 ] 20.9218 23.2465 1.2589

B.2

Phase Shifts Obtained by the Infinity Norm Algorithm 2

1.8

1.6

φk [−] (×π)

1.4

1.2

1

0.8

0.6

0.4

0.2

0

2

4

6

8

10

12

14

16

18

20

2p [−] Figure B.1: Convergence of the optimal phase shifts for a 25 sine multisine with flat, equally spaced spectrum between 0.01 [Hz] and 25 [Hz], using the infinity norm algoritm (Pintelon and Schoukens, 2001).

Table B.2: Phase shifts obtained using the infinity norm to p = 10. φ1 = 1.5706 φ6 = 1.8201 φ11 = 0.6606 φ2 = 0.1425 φ7 = 1.0641 φ12 = 1.4765 φ3 = 1.2977 φ8 = 0.0000 φ13 = 1.7126 φ4 = 1.7042 φ9 = 0.7113 φ14 = 1.8925 φ5 = 1.4277 φ10 = 0.6608 φ15 = 1.5232

algorithm φk [−] (×π) with Nf = 25 up φ16 φ17 φ18 φ19 φ20

= 0.8378 = 1.9196 = 1.9680 = 0.0724 = 1.2242

φ21 φ22 φ23 φ24 φ25

= 0.1689 = 0.0000 = 1.3600 = 0.1597 = 0.6171

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

70

B.3

Nonlinear Force-Displacement Characteristics

Table B.3: Phase shifts obtained using the infinity norm to p = 2. φ1 = 0.5160 φ11 = 2.9592 φ21 = 2.0268 φ2 = 1.0146 φ12 = 0.8852 φ22 = 0.1898 φ3 = 1.2792 φ13 = 1.2443 φ23 = 1.3690 φ4 = 0.1630 φ14 = 1.6920 φ24 = 0.5438 φ5 = 0.2704 φ15 = 1.3122 φ25 = 1.6525 φ6 = 0.6175 φ16 = 1.5676 φ26 = 0.7727 φ7 = 1.8581 φ17 = 0.7970 φ27 = 0.3312 φ8 = 0.4419 φ18 = 0.4583 φ28 = 0.1863 φ9 = 1.4532 φ19 = 0.0768 φ29 = 1.2465 φ10 = 0.3311 φ20 = 0.1754 φ30 = 0.9476

B.3

φ31 φ32 φ33 φ34 φ35 φ36 φ37 φ38 φ39 φ40

= 0.3167 = 1.2186 = 1.5124 = 1.1894 = 0.7794 = 0.3178 = 0.8564 = 0.7474 = 1.0013 = 1.0380

φ41 φ42 φ43 φ44 φ45 φ46 φ47 φ48 φ49 φ50

= 0.0072 = 0.6064 = 0.8849 = 1.7063 = 0.9052 = 0.5380 = 0.8555 = 1.7034 = 1.8075 = 0.4661

Nonlinear Force-Displacement Characteristics Table B.4: Coefficients ρij of the i=1 j=0 0.000234722612437 j=1 0.091796394850297 j = 2 −0.005071505003798 j=3 0.001755823061963 j = 4 −0.000495321921394 j=5 0.000455358409592

B.4

algorithm φk [−] (×π) with Nf = 50 up

force-displacement characteristics in the set-up. i=2 i=3 0.005243242423923 −0.003305403062789 0.084268742935300 0.680504615284765 −0.052699540025315 0.089803957869115 0.015739329106770 0.203236186622692 0.063153998888216 −0.064473933604213 0.020271796321899 0.017105175330557

Additional Figures

0.5 Measurement

1

Simulation

Measurement

Simulation

ξ2 [−]

ξ2 [−]

0.5

0

0 −0.5 −1

−0.5

−1.5 0

10

20

30

40

50

60

0

500

0

500

1000

1500

1000

1500

0.4

0.1

0.3

e2 [−]

e2 [−]

0.05

0

−0.05

0.2 0.1 0 −0.1

0

10

20

30

τ [−] (a)

40

50

60

τ [−] (b)

Figure B.2: Measured and simulated response to (a) an input signal v(t) and (b) a step signal with increasing amplitude (red oscillator), (Top) Response, (Bottom) Error: er = ξr,s − ξr,m , where ξr,s denotes simulation results and ξr,m the corresponding measured response.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

C Technical Specifications

C

71

Technical Specifications

This appendix discusses the main technical issues present in the set-up. First, a scheme showing the correct configuration of the experimental set-up and additional components is introduced as well as a detailed schematic of the emergency brake system. Next, measurements of the masses of the components of the set-up and the measured eigenfrequencies of the different parts are provided and the selection of at data acquisition system and the calibration of the voice coils is discussed. Moreover, a detailed discussion of the damping phenomena that are present in the set up is presented and finally the specifications of the sensors and linear motors are presented.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

DAC1 DAC2

ADC1 ADC2

From blue oscillator

TUeDACs BLN SN 0518039

From red oscillator

To beam

To red oscillator

To blue oscillator

Signal Hub

Amplifier 35-1219

Amplifier 35-1214

Amplifier 35-1218

230 V

230 V

Emergency Stop [Mechanical & Power / Heat]

STOP

C.1

PC

DAC1 DAC2

ADC1 ADC2

TUeDACs BLN SN 0518004

All components are grounded to a common ground.

Sensor power (24 V)

Mechanical safety loop (24 V)

Actuator signal (0-5V & 24V, 8W aft. amp.)

Power (230 V), safety protected

Power (230 V), no safety

From beam

C.1

Set-up

72 Connection Scheme Set-Up

Connection Scheme Set-Up

Figure C.1: Configuration of the experimental set-up and additional components.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

C.2

Safety System

C.2

Safety System

Figure C.2: Electrical scheme of the safety system accompanying the experimental set-up.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

73

74

C.3 Masses and Eigenfrequencies of Set-up

C.3

Masses and Eigenfrequencies of Set-up

Table C.1: Measured weight of different parts Part Mass [g] Comment Main beam 2344.75 Horizontal leaf spring (big) 341.84 323.48 Vertical leaf spring Oscillator frame (top) 363.42 140.50 Oscillator frame (bottom) Oscillator frame (lever) 42.53 Oscillator body 208.85 Negative stiffness mechanism oscillator 43.19 Actuator (house) 326.99 Actuator (coil) 56.03 1.238 Standard leaf spring

Table C.2: Measured eigenfrequencies of the beam and Part Main beam Oscillator (no sensor and actuator)

C.4

of the set-up.

50% moving mass 50% moving mass Moves with beam 200% of oscillator motion 150% of the oscillator motion

Moves with beam Moves with oscillator

oscillator without actuators and sensors. Eigenfrequency [Hz] 1.92 3.64

Data Acquisition Systems

Three of the Data Acquisition System (DACS) that have been considered are D-space, Quansarcards and the TUeDACS. The latter is a generic term for a collection of DACS. Taking into account the fact that the system specifications might change in the future, the main aspects considered were: 1. Measurement speed 2. Price 3. Modular applicable (in case the number of sensors / actuator changes in the future) Both D-space and the Quansar cart are rather expensive compared to the TUeDACS. Therefore attention is focussed on the TUeDACS, which yielded five possible options: 1. TUeDACS/1 Advanced Quadrature /analog / digital interface 2. TUeDACS/1 Quadrature /analog / digital interface 3. TUeDACS/6 Analog Signal Generator 4. TUeDACS/3 1 Mhz 12 bit Parallel Sampling ADC 5. TUeDACS/1 MicroGiant Options 1 and 2 are still obtainable, but have been replaced by the more advanced MicroGiant system (option 5). Therefore the combination of option 3 (for actuation control) and 4 (for sampling sensor data) and the MicroGiant remain as final options. Although the combination of 3 and 4 provides the possibility to control four incoming and outgoing signals, this option would require the purchase of two separate systems. The MicroGiant system, on the other hand, is already present in the D&C laboratory and is modular applicable. Since each MicroGiant allows for two incoming analog signals and two outgoing analog signals the present set-up requires two MicroGiant systems, which can be connected to one PC via a standard USB hub. Futhermore, the MicroGiant system is capable of measuring with frequencies up to 8 kHz (1 kHz without data loss, 4 kHz without Huygens Synchronization in Various Dynamical Systems. Experimental Results.

C.5

Relative Calibration of the Voice Coil

75

technical modification). Summarizing, the MicroGiant is the least expensive option that has been considered. Furthermore, the possible measurement speed meets the present needs and modular applicability allows for future modifications to the set-up.

C.5

Relative Calibration of the Voice Coil

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

ξ2 [−]

ξ1 [−]

In order to compensate for differences between the different actuator / amplifier combinations within the set up, each of the oscillators has been fixed to the fixed world sequentially. Next an identical , but sign reversed, sine wave is supplied to actuator acting on the beam and to the actuator acting on the red or blue actuator. The resulting motion will be zero if the actuator strengths (actuator and amplifier) are equal. By multiplying the sine wave applied to the red or blue actuator with a gain γ ∈ [0 2] the nett force Fbeam − Fblue/red is varied slowly. The results of these experiments are shown in Figure C.3. As expected the amplitude of the oscillation reaches a minimum if the blue / red and black actuator are in equilibrium. This occurs at γb = 1.078 [−] and γr = 1.109 [−]. Therefore, it can be concluded that the blue actuator is 7.8% ’weaker’ than the black one and the red actuator is 10.9% weaker as well. In future experiments this discrepancy is compensated by supplying the appropriate gain to the output signals that is supplied to the actuators.

0

−0.1

0

−0.1

−0.2

−0.2

−0.3

−0.3

−0.4

0

0.2

0.4

0.6

0.8

1

1.2

γb [−] (a)

1.4

1.6

1.8

2

−0.4

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

γr [−] (b)

Figure C.3: Results of the relative actuator calibration experiment. (a) Blue oscillator, (b) Red oscillator.

C.6

Damping Analysis

This Section discusses the presence and possible causes for the damping that is observed in the system. First, the possible causes for viscous damping are investigated and damping due to air friction is specifically ruled out as a candidate for this type of friction. Finally, the effects of hysteresis and dry friction / stick-slip are identified and methods are proposed to minimize these influences. Please note the use of non-dimensionless parameters in this Section, in order to provide easily applicable results. C.6.1

Viscous Damping

One of the technical issues with the set-up is the fact that the voice coil actuators (linear motors) appear to damp the motion of the oscillators much more than was expected. This damping appears to be a linear function of the velocity of the coils. However, the laws of physics tell us that if a coil is Huygens Synchronization in Various Dynamical Systems. Experimental Results.

76

C.6

Damping Analysis

not short circuited nor connected to an amplifier there should be no force applied to the coil when moving through a magnetic field. Therefore three possible causes for the fact that a (relatively strong) damping force is present in the system, are suggested: 1. A partly internally short circuited coil. 2. Resistance from the air moving in and out of the coil chamber through the narrow slit. 3. Current running though parts of the linear motor, other than the coil. The first option is highly unlikely because all three motors in the set-up show the same damping behavior. It is therefore disregarded for now. The second option will be discussed in more detail in the following paragraph. Consider the schematic representation of a linear motor / voice coil in Figure C.4. The coil is moving downwards with a velocity vc and air from a chamber with radius ρ is pushed out of the chamber through a narrow slit of width t. Note that the system depicted in Figure C.4 is rotation symmetric around the vertical axis. Next, the (approximate) drag force Fd exerted on the coil as a result of the air flow through the narrow slit is computed.

vc

Coil

ρ

t 2

t 2

va (ξ) ℓ ξ

Stator

Figure C.4: Schematic representation of a voice coil actuator.

Assuming constant pressure throughout the system, the average velocity of the air in the slit v¯a is obtained from conservation of volume: v¯a = vc

Achamber Aslit

(C.1)

However, the velocity profile of the air in the slit is a Poiseuille flow and follows a parabolic profile as depicted in Figure C.4. The Poiseuille profile may be described as a function of the spatial Huygens Synchronization in Various Dynamical Systems. Experimental Results.

C.6

Damping Analysis

77

coordinate ξ perpendicular to the slit walls as: v(ξ) = αξ 2 + βξ + γ

(C.2)

The coefficients α, β, γ ∈ R are obtained from three conditions: a 1. Symmetry of the Poiseuille profile: dv =0 dξ ξ=0

t

t

2. Conservation of volume (transfer):

R2

va (ξ) dξ =

− 2t

R2

v¯a dξ

− 2t

3. No-slip condition at the wall: v(ξ = ± 2t ) = 0

v ¯a Condition 1 immediately yields β = 0 and conditions 2 and 3 yield α = − 12 11 t2 and γ = yields the velocity profile: !  2 12 ξ va (ξ) = −¯ va −1 11 t

12 ¯a . 11 v

This (C.3)

The drag force exerted on the coil can now be approximated by: Fd = ηAwall ∇va (ξ)|ξ= t , 2

(C.4)

where η = 13.30 · 10−6 m2 s−1 is the kinematic viscosity of air at room temperature (20 ◦ C) and Awall is the area of the coil-wall in the slit. Furthermore: ∇va (ξ)|ξ= t = 2

12 v¯a . 11 t

(C.5)

Given that ρ = 3 · 10−2 m, t = 0.5 · 10−3 m, ℓ = 4.83 · 10−2 m and vc reaches a maximum of approximately vc = 0.15 ms−1 and Achamber Aslit

=

πρ2

=

2  2 !  t t = 2ρt, − ρ− π ρ+ 2 2

yielding, v¯a = and

ρ vc = 4.5 ms−1 , 2t

12 v¯a = 1.0 · 104 ms−2 . 11 t m2 :

∇va (ξ)|ξ=t = Therefore, with Awall = 2πρℓ = 9.1 · 10−3

Fd = ηAwall ∇va (ξ)|ξ=t = 1.2 · 10−3 N

(C.6) (C.7)

(C.8)

Given that the mass of an oscillator is approximately mo = 0.6 kg, this drag force will result in a −3 maximal deceleration of ao = − 1.2·10 = 2 · 10−3 ms−2 , which can by no means be responsible for 0.6 the damping observed in the system. Please note that the actual influence of air damping will be much less since the velocity used is the maximal velocity reached by the system, which is only obtained for a small fraction of the actual motion. Furthermore, the area used in (C.8), that is actual causing the drag, will be much smaller during most of the motion since only a fraction of ℓ will be located in the slot. Concluding, the fraction of damping that is caused by airflow through the slits in the motor is small and will be neglected. This leaves the final option ’Current running though parts of the linear motor, other than the coil’ as the most probable cause of damping. Checking with the manufacturer of the motors support Huygens Synchronization in Various Dynamical Systems. Experimental Results.

78

C.6

Damping Analysis

this conclusion since the engines used in the set-up have aluminium covers that allow a current to flow, even if the coils are disconnected. Engines with polymer covers exist, but have regretfully not been used here. Now that the cause for the damping is known, the damping itself does not cause a particular problem, since the damping caused by back emf is a linear function of velocity and may easily be compensated by an appropriate feed forward. C.6.2

Hysteresis and Dry Friction

With the main cause for the damping identified, the only remaining issue is the shift in equilibrium position that is observed in measurements. Two candidates for the cause of this issue are: 1. Hysteresis 2. Dry friction. Both of these effects may be observed when exiting the individual oscillators and beam by a slow saw tooth in force. If no hysteresis is present the mass should follow the exact same trajectory moving from left to right as moving back from right to left. Dry friction will become apparent from jumps or steps in the measurements, rather than a smooth curve. An example of such measurement, performed with the red oscillator is presented in Figure C.5. In this experiment the oscillator was moved from left to right over it’s entire stroke in 60 seconds. The experiment was repeated five times. From Figure C.5 it is observed that in this case both hysteresis and dry friction appear to be present in the subsystem. Clearly the oscillator follows a different trajectory moving ’upwards’ then when moving ’downwards’. Here δ represents the maximum change in equilibrium position that may be observed as a result from both hysteresis and dry friction. 0.1

0.08

0.06

0.04

slip

F [V]

F [V]

0.02

0

-0.02

-0.04

stick

δ

-0.06

-0.08

-0.1 -10

-8

-6

-4

-2

0

2

4

6

8

x [mm] (a)

x [mm] (b)

Figure C.5: Stick-slip and hysteresis in red oscillator.

Although the experiment presented in Figure C.5 indicates that friction is indeed present in the system (and provides information about where the problem is located) it does not tell exactly what role hysteresis plays. Experiments with smaller force maxima which make sure the dry friction is never reached show that even then δ 6= 0, thus hysteresis is present in addition to the observed friction. However δ as a result of friction is much larger than δ caused by hysteresis. To further quantify this reasoning 10 sequential experiments have been performed on each oscillator and the beam. In these experiments the amplitude of the saw tooth force was linearly varied between zero and the maximal allowable force (0.42 V for the beam, and 0.105 V for the oscillators). For each experiment δ is computed and plotted versus the distance traveled over half a saw tooth, i.e. half Huygens Synchronization in Various Dynamical Systems. Experimental Results.

C.6

Damping Analysis

79

the difference between the maximum and minimum measured position). The results are depicted in Figure C.6. 1.4 Blue Oscillator Red Oscillator Beam 1.2

1

δ [mm]

0.8

0.6

0.4

0.2

0

0

2

4

6

8

10

12

ϑ [mm] Figure C.6: Hysteresis and friction influences as a function of the amplitude of oscillation. Here δ represents the uncertainty in equilibrium position after traveling with an amplitude ϑ.

From Figure C.6 and the individual measurements the following may be concluded: Red Oscillator The red oscillator contains a dry friction element at about 6 mm from its equilibrium position. Since the cause for this friction is located in either the engine or the sensor the problem can not be solved directly. Instead the part of the stroke of the red oscillator that shows this behavior x ≥ 5.8 mm will no longer be used in experiments. At this position a physical barrier has been created. Blue Oscillator The blue oscillator only contains hysteresis, which does not cause a significant problem at this time. Beam The beam also contains hysteresis, but the effects are much stronger than in the oscillators for larger movements. However, in normal operation mode the movements of the beam stay confined to small oscillations and this hysteresis is therefore not expected to cause significant problems.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

80

C.7

C.7

Linear Variable Differential Transformer

Linear Variable Differential Transformer

This appendix provides the manufacturers specifications regarding the LVDT position sensors (measurement range (MB): 0 ... 25 mm) used in the set-up. Copyright and responsibility remain with the manufacturer (WayCon Positionsmesstechnik GmbH, [email protected]). Source: www.waycon.de

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

LVDT

Induktiver Miniatur Wegaufnehmer

Serie SM ø12 mm

• Messbereiche 2...200 mm • Linearität 0,3% (0,20% auf Anfrage) • ø12 mm, Spann-ø 8mm h6 • Ausgang: AC, 0...10 V, 0...5 V, 4...20 mA • Mit Extern- oder integrierter Kabelelektronik • Wiederholgenauigkeit bis zu 1,5 µm • Gehäuse aus vernickeltem Stahl • Betriebstemperatur -40...+120°C (150°C optional) • Sonderbauformen für Hydraulik- und Druckbereich

04.05.06

-2Technische Daten Sensor Messbereiche [mm] Linearität Ausführung Lagerwerkstoff Schutzklasse Vibrationsfestigkeit DIN IEC68T2-6 Schockfestigkeit DIN IEC68T2-27 Nennspeisespannung/Frequenz Speise-Frequenzbereich Temperaturbereich Befestigung Anschluss Gehäuse Kabel -PVC (standard) -PTFE (optional) -PUR (optional) max. zul. Kabellänge Federtaster (bis MB ±50mm) Federkraft typ. min./max. [N] max. Bewegungsfrequenz d. Tastspitze bei 1 mm Hub [Hz] Federkonstante Zugfeder [N/mm] Gewicht (ohne Kabel) [g] Lebensdauer freier Anker / Stößel max. Beschleunigung d. Ankers/Stößels Lebensdauer Gewicht (ohne Kabel) [g]

Elektronik Ausgangssignal

0...2 0...5 0...10 0...25 0...50 0...100 0...200 0,3% (0,20% optional) Taster (bis MB 0...50), freier Anker, Stößel mit/ohne Lagerung Phosphor-Bronze IP65 oder IP68 / 10 bar 10 G 200 G / 2 ms 3 Veff / 3 kHz 2...10 kHz -40...+120°C (150° optional, H-Option) ø8 mm h6 Spanndurchmesser oder ø12 mm Klemmblöcke Kabelanschluss 4-pol. PVC-Schirm-Kabel bzw. PTFE-Kabel (Option H) oder M12-Steckeranschluss, verschraubbar Vernickelter Stahl ø4,7 mm, 0,16 mm², 2 verdrillte PVC-isolierte Paare ø3,7 mm, 0,24 mm², max. Temp. 205°C ø3,9 mm, 0,14 mm², halogenfrei, hochflexibel 100 m zwischen Sensor und Externelektronik IMA 0,5/0,6 55

0,6/0,7 50

0,016 0,011 ca. 48 ca. 55 >10 Mio. Zyklen 100 G unendlich ca. 36 ca. 42

0,6/0,7 50

0,7/0,75 35

0,007 ca. 72

0,004 ca. 105

ca. 47

ca. 59

0,75/0,8 20

ca. 85

Sensorversorgung

IMA Externelektronik (Schaltschrankeinbau) 0...20 mA, 4...20 mA (Last 5 kOhm) 0...10 V, ±10 V (Last >10 kOhm) Nullpunkt 150 ppm/°C, Endwert 400 ppm/°C < 20m Veff 300 Hz/-3 dB (Butterworth 5‘ter Ordnung) Offset ±20%, Verstärkung ±50% > 1 GOhm bei 500 VDC Versorgung Ausgang 500 VDC 24 VDC (18..36 V) oder 15 VDC (9..18 V) 10 kOhm) 460 ppm/°C < 20m Veff 24 VDC (18..36 V) od. 15 VDC (9..18 V) 65 mA (24 VDC), 140 mA (12 VDC) 3,0 Veff (15...26V-Versorgung) 2,4 Veff (12...20V-Versorgung) 0... +60°C -20... +80°C Aluminium trovalisiert keine

120 (10 V/20 mA) 100 80 Ausgangs60 signal [%] 40 elektrischer Messbereich

20

-20

0 -20

20

40

Einfahren

60

80

100

Ausfahren

Messbereich [%]

120

-3Technische Abmessung

Messbereich (MB) [mm]

Gehäuselänge Kabel axial [mm]

0...2 0...5 0...10 0...25 0...50 0...100 0...200

58 64 74 104 154 254 454

Gehäuselänge Kabel radial [mm] 68 78 84 114 164 264 464

Gehäuselänge Stecker M12 [mm] 67 73 83 113 163 263 463

Ausführung mit freiem Anker oder Stößel

Stößel gelagert

Federtaster (bis max. MB 0...50)

Bitte beachten Sie, dass der angegebene Anhub und Endhub (siehe Kreis) Richtwerte sind. Bei Kalibrierung des Sensors vor Auslieferung wird auf maximale Linearität geachtet. Ein Abgleich auf die exakten Werte ist jedoch möglich. Bitte geben Sie dies bei Bestellung gesondert an.

Ankerlänge [mm] 22 25 30 45 70 120 220

Stößellänge [mm] 54 60 70 100 150 250 450

-4Ausgänge (optional) Geräte mit axialem Kabelausgang besitzen die kleinste Gehäuselänge. Der Biegeradius bei Verlegung sollte den dreifachen Kabeldurchmesser nicht unterschreiten. Die Standard-Kabellänge beträgt 2 m.

Kabelausgang axial

Geräte mit der Option H für Temperaturen bis 150°C besitzen eine PGVerschraubung mit SW14. Standard

Option H

Geräte mit radialem Kabelausgang besitzen standardmäßig eine Durchgangsbohrung. Bitte verwenden Sie diese Variante für die Verwendung unter starker Schmutzeinwirkung. Durch die Bewegung des Stößels wird die Verschmutzung aus dem Sensor nach hinten abtransportiert. Die Standard-Kabellänge beträgt 2 m.

Kabelausgang radial

Standard

optional

Für normale Anwendungen kann der Sensor auf Wunsch auch rückseitig verschlossen geliefert werden (ohne Aufpreis). Bitte geben Sie das bei der Bestellung gesondert an. Die Kombination der Optionen H (150°C) und KR (Kabelausgang radial) ist nicht möglich.

Steckerausgang (Kabel mit geradem oder Winkelstecker)

Für Geräte mit Steckerausgang muss das Kabel gesondert bestellt werden. Hierbei stehen Kabel mit geradem Stecker oder mit Winkelstecker zur Verfügung. Der Stecker wird durch Verschraubung (M12) gegen versehentliches Abziehen gesichert. Die Kabellängen betragen 2/5/10 m. Die Steckverbindung hat Schutzklasse IP65. Die gesamte Sensorlänge mit Winkelstecker beträgt: Gehäuselänge Stecker M12 (siehe Tabelle) + 20 mm (Winkelstecker) Gehäuselänge Stecker M12 (siehe Tabelle) + 37mm (gerader Stecker)

Folgende Sonderbauformen erhalten Sie auf Wunsch: - Sondermessbereiche (z.B. X mm) - Druckdichte Geräte mit Einbauflansch - Geräte für Unterwassereinsatz - Geräte mit verkürztem Gehäuse Steckerbelegung

Einstellung von Nullpunkt und Verstärkung der Elektronik Bitte beachten Sie, dass sich Nullpunkt und Verstärkung bei großen Leitungslängen zwischen Sensor und Elektronik verschieben können. Installieren Sie daher den Sensor mit der erforderlichen Leitungslänge zur Elektronik und nehmen Sie dann die Einstellung von Nullpunkt und Verstärkung vor. 1. Stößel in Nullage - Offset einstellen. Verfahren Sie den Sensor in den Nullpunkt des Messbereiches Stellen Sie das Offset-Potentiometer auf 0 mA bzw. 0 V Ausgangssignal ein. 2. Stößel in Endlage - Verstärkung einstellen. Verfahren Sie den Sensor auf den mechanischen Endpunkt (Stößel ausgefahren). Stellen Sie das Verstärkungs-Potentiometer auf 16 mA / 10 V / 5 V Ausgangssignal ein. 3. Offset einstellen (4...20 mA Ausgang). Stellen Sie mit dem Offset-Potentiometer 20 mA (+4 mA) das Ausgangssignal ein. Hinweis zur Richtungsumkehr: Sollten Sie ein invertiertes Ausgangssignal benötigen (20...4 mA / 10...0 V / 5...0 V), so tauschen Sie die Klemmen 6 und 8 (Sekundärspule) an der Externelektronik.

-5AC-Ausgang blau rot grün weiss

Kabelbelegung: weiss (5): Primär 2 grün/schwarz (6): Sekundär 2 rot/braun (9): Primär 1 blau (8):Sekundär 1

Kabelbelegung für PTFE-Leitung: weiss (5): Primär 2 grün (6): Sekundär 2 gelb (9): Primär 1 braun (8): Sekundär 1

Kalbelelektronik KAB

Kabellänge Sensor-Elektronik 1 m, 4 m oder 9 m

Kabellänge 1 m

Sensorseite

Standardgemäss befindet sich die Kabelelektronik 1 m vor Kabelende. Auf Wunsch ist diese jedoch an beliebiger Stelle konfektionierbar. Bitte bei Bestellung angeben.

Kabelbelegung: braun/rot: blau: schwarz/grün: weiss:

Anschlussseite

Versorgung V+ GND Ausg. GND Ausg. Signal

Kabelbelegung für PTFE-Leitung: gelb: Versorgung V+ braun: GND grün: Ausg. GND weiss: Ausg. Signal

Externelektronik IMA Abmessungen: 1 2 3 Erde* Gnd (24 VDC) 24 VDC

Phase Ampl. Verst. Offset 4 5 6 7 8 910

n.c. Primär 2 Sekundär 2 Schirm* Sekundär 1 Primär 1 n.c.

Schirm* Signalausgang Gnd (Signal)

11 12 13

Externelektronik IMA (für DIN-Schienenmontage)

* Die Klemmen 1, 7 u. 13 sind geräteintern verbunden

Anschluss Die Externelektronik IMA2-LVDT ist für den Schaltschrankeinbau (DIN-Schienenmontage) konzipiert. Der Anschluß für den Wegaufnehmer ist als Stecker mit Schraubklemmen ausgeführt.

Bei schwierigen EMV-Bedingungen besteht die Möglichkeit, die Elektronik bis zu 100 m entfernt in einem Schaltschrank unterzubringen. Für die Verdrahtung zwischen Sensor und Externelektronik ist ein paarweise verdrilltes Kabel (Twin-Twisted-Pair, 4-Adrig, Mindestquerschnitt 0,5 mm²) mit Einfach- oder Doppel-Abschirmung zu verwenden. Vorzugsweise ist der Schirm im Schaltschrank nahe der Elektronik zu erden. Das Sensorgehäuse wird über das Maschinenchassis geerdet. Die Kabellänge sollte wegen der Störbeeinflussung 100 m nicht überschreiten.

Wegaufnehmer

Klemmkasten

Verbindungskabel

Elektronik Installation im Schaltschrank Versorgung

prim. sek. Schirm

Signalausgang

-6Bestellcode SM Messbereich [mm] 0... 2 0....5 0...10 0...25 0...50 0...100 (nicht für Federtaster) 0...200 (nicht für Federtaster)

2 5 10 25 50 100 200

Ausführung freier Anker Stößel Stößel gelagert Federtaster (bis max. 0...50 mm)

A S SG T

Externelektronik Kabelelektronik

IP68 H L20 SA KA KR

IMA-3A KAB

Spannungsversorgung 12 VDC (nicht 10/±5V-Ausgang) 24 VDC

IP65 IP68 (nur mit PTFE-Kabel) Temp. 150°C verbesserte Linearität 0,20% (nicht Opt. H) Stecker M12 Axialer Kabelausgang (nicht Opt. H) Radialer Kabelausgang

Ausgang 0...20 mA 4...20 mA 0...10 V 0...5 V ±5V ± 10 V

020A 420A 10V 5V ±5V ±10V

12 V 24 V

Preise SM2

0...2 mm

184 €

SM5

0...5 mm

205 €

SM10

0...10 mm

225 €

SM25

0...25 mm

243 €

SM50

0...50 mm

250 €

SM100

0...100 mm

297 €

SM200

0...200 mm

355 €

Optionen: A S SG T IP68 H L20

Anschlusskabel: freier Anker Stößel Stößel gelagert Federtaster Wasserdicht bis 10 Bar erhöhter Temp.-Bereich 150°C verbesserte Linearität 0,20% (auf Anfrage)

16 € 41 € 51 € 78 € 46 €

integrierte Kabelelektronik Schaltschrankelektronik

174 € 264 €

100 €

Elektronik: KAB IMA

Kabel mit geradem Stecker M12 (SA) K4P2M-S 2m 14 € K4P5M-S 5m 17 € K4P10M-S 10 m 22 € Kabel mit Winkelstecker M12 (SA) K4P2M-SW 2m 14 € K4P5M-SW 5m 17 € K4P10M-SW 10 m 22 € Festes Anschlusskabel (2,0 m Standard, KA, KR): je weiterer Meter PVC-Kabel 6 € /m je weiterer Meter PUR-Kabel 6 € /m je weiterer Meter PTFE-Kabel (-H) 10 € /m Diese Daten können jederzeit ohne Vorankündigung geändert werden

WayCon Positionsmesstechnik GmbH e-mail: [email protected] internet: www.waycon.de

Head Office Mehlbeerenstr. 4 82024 Taufkirchen Tel. +49 (0)89 67 97 13-0 Fax +49 (0)89 67 97 13-250

Office Köln Kierberger Str. 24 50321 Brühl Tel. +49 (0)2232 56 79 44 Fax +49 (0)2232 56 79 45

C.8

C.8

Linear Motor (Voice Coil Actuators)

87

Linear Motor (Voice Coil Actuators)

This appendix provides the manufacturers specifications regarding the voice coil actuators (type NCC10-15-023-1X) used in the set-up. Copyright and responsibility remain with the manufacturer (H2W Technologies, Inc.). Source: www.h2wtech.de

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

88

C.8 Linear Motor (Voice Coil Actuators)

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

.500

These drawings and specifications are the property of H2W Technologies, Inc. They are issued in confidence and shall not be reproduced, copied, or used without written permission from H2W Technologies, Inc.

1.000

1.500

UNLESS SPECIFIED OTHERWISE: All dimensions are in inches Standard Tolerances are as follows .XXX .005 ANGLES ±1° .XX .010 FILLETS AND .X .020 CORNERS .010 Remove All Burrs and Sharp Edges

2.160

1.910

OOG

DRAWN

FINISH

MATERIAL

6-32 x .20 Dp Mtg Holes (3 Pl)

3-29-02

DATE

FGW

APPROVED

1.230

.600

3-29-02

DATE

DWG #

TITLE

NCC10-15-023-1X 1.00" 0.010" per side 56 grams 375 grams 7.3 ohms 0.81 LBS/(W att)^1/2 2.30 LBS 6.90 LBS 8 W ATTS

.300

NCC10-15-023-1X ECN REV 30-0114 B 1393

28310-C Ave Crocker Valencia, CA 91355 USA Tel: (661)-702-9346 Fax: (661)-702-9348 www.h2wtech.com

H2W Technologies, Inc.

SPECIFICATIONS Motor P/N Stroke Radial Clearance Moving Mass Total Mass Resistance Force Constant Continuous Force Peak Force Power In @ 100% Duty

Power leads

6-32 x.20 Dp Mtg Holes (2 Pl)

90

C.8 Linear Motor (Voice Coil Actuators)

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

D Related Papers and Conferences

D

91

Related Papers and Conferences

In this section the ENOC 2008 paper related to this project is supplied. This paper is presented at the ENOC 2008 conference in St. Petersburg.

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

92

D Related Papers and Conferences

Huygens Synchronization in Various Dynamical Systems. Experimental Results.

ENOC-2008, Saint Petersburg, Russia, June, 30–July, 4 2008

SYNCHRONIZATION BETWEEN COUPLED OSCILLATORS: AN EXPERIMENTAL APPROACH

David Rijlaarsdam

Alexander Yu. Pogromsky

Henk Nijmeijer

Dept. of Mechanical Engineering Eindhoven University of Technology The Netherlands [email protected]

Dept. of Mechanical Engineering Eindhoven University of Technology The Netherlands [email protected]

Dept. of Mechanical Engineering Eindhoven University of Technology The Netherlands [email protected]

Abstract We present an experimental set-up that allows to study both controlled and uncontrolled synchronization between a variety of different oscillators. Two experiments are presented where uncontrolled synchronization between two types of identical oscillators is investigated. First, uncontrolled synchronization between two Duffing oscillators is investigated and second, uncontrolled synchronization between two coupled rotating elements is discussed. In addition to experimental results we provide both analytical and numerical results that support the experimental results.

Key words Synchronization, Experiment, Duffing oscillator, Huygens experiments

1 Introduction In the 17th century the Dutch scientist Christiaan Huygens observed a peculiar phenomenon when two pendula clocks, mounted on a common frame, seemed to ’sympathize’ as he described it. What he observed was that both clocks adjusted their rhythm towards antiphase synchronized motion. This effect is now known as frequency or Huygens synchronization and is caused by weak interaction between the clocks due to small displacements of the connecting frame. In (Bennett et al., 2002; Pantaleone, 2002; Senator, 2006; Kuznetsov et al., 2007) an extended analysis of this phenomenon is presented. In (Oud et al., 2006) the authors present an experimental study of Huygens synchronization and finally, in (Pogromsky et al., 2003; Pogromsky et al., 2006) a study of the uncontrolled as well as the controlled Huygens experiment is presented. In this paper we present an experimental set-up (Tillaart, 2006) that allows to study both controlled and uncontrolled synchronization between a variety of different oscillators. In section 2 the set-up is introduced and we present the dynamical properties of the system.

Furthermore, we present the means by which we are able to modify these properties to represent a variety of different oscillators. Next, in section 4, we present an experiment of the synchronization of two Duffing oscillators. We analyze the stability of the synchronization manifold and continue with numerical and experimental results. Section 5 presents an experiment where the set-up is adjusted to model two rotating eccentric discs which are coupled through a third disc mounted on a common axis. Conclusions and future research are presented in section 6. 2 Experimental Set-up In order to experimentally study synchronization between coupled oscillators a set-up consisting of two oscillators, mounted on a common frame has been developed (see figure 1 and 2). The parameters of pri-

Figure 1.

Photograph of the set-up.

mary interest are presented in table 1. The set-up contains tree actuators and position sensors on all degrees of freedom. Furthermore, although the masses of the

oscillators (m) are fixed, the mass of the connecting beam (M ) may be varied by a factor 10. This allows for mechanical adjustment of the coupling strength.

Table 1.

Parameters in experimental set-up.

Oscillator 1

Oscillator 2

Frame / beam

m

m

M

Stiffness

κ1 (·)

κ2 (·)

κ3 (·)

Damping

β1 (·)

β2 (·)

β3 (·)

Mass

x3 κ3 β3

κ1 β1

F3

F1

x2

x1

Figure 2.

κ2 β2

m M

3 Definition of Synchronization Before continuing with the experimental and analytical results the notion of synchronization should be defined in more detail. Due to the large amount of phenomena that seems to be gathered under the term synchronization, it is often difficult to correctly define synchronization. In (Pikovsky et al., 2001) the authors introduce the concept of synchronization as:

m

F2

Schematic representation of the set-up.

A schematic representation of the set-up is depicted in figure 2 and the equations of motion are: m¨ x1 = −κ1 (x1 − x3 ) − β1 (x˙ 1 − x˙ 3 ) + F1 m¨ x2 = −κ2 (x2 − x3 ) − β2 (x˙ 2 − x˙ 3 ) + F2

(2.1) (2.2)

Mx ¨3 = κ1 (x1 − x3 ) + κ2 (x2 − x3 ) (2.3) − κ3 (x3 ) + β1 (x˙ 1 − x˙ 3 ) + β2 (x˙ 2 − x˙ 3 ) − β3 (x˙ 3 ) + F

3

2.1 Adjustment of the Systems’ Properties In order to experiment with different types of oscillators, the derived properties (stiffness and damping) are adjusted. Note that, since we know the damping and stiffness present in the system and we can fully measure the state of the system, we may adjust these properties, using actuators, to represent any dynamics we want. This allows modeling of different types of springs (linear, cubic), gravity (pendula) and any other desired effect within the limits of the hardware. In the next part of this paper we present two examples of this type of modulation. The system is first adapted to analyze synchronization between Duffing oscillators and secondly to analyze the synchronizing dynamics of two coupled rotating eccentric discs under the influence of gravity.

− F1 − F2 ,

where m, M ∈ R>0 and xi ∈ Di ⊂ R, i = 1, 2, 3 are the masses and displacements of the oscillators and the beam respectively. Functions κi : R 7→ R, βi : R 7→ R describe the stiffness and damping characteristics present in the system. Fi are the actuator forces that may be determined such that the experimental setup models a large variety of different dynamical systems (see 2.1). The stiffness and damping in the system are found to be very well approximated by:

κi (qi ) =

5 X

kij q j

(2.4)

j=1

βi (q˙i ) = bi q˙i ,

(2.5)

where q1 = x1 − x3 , q2 = x2 − x3 and q3 = x3 . The values of kij and bi ∀ i = 1, 2, 3 have been experimentally obtained and will be used to modify the systems’ properties in the sequel.

Synchronization is the adjustment of rhythms of oscillating objects due to their weak interaction. Although the above concept provides an insightful idea of synchronization a more rigorous definition is provided in (Blekhman et al., 1997): Definition 3.1: (Asymptotic Synchronization). Given k systems with state xi ∈ Xi and output yi ∈ Yi , i = 1, . . . , k and given ℓ functionals gj : Y1 ×. . .× Yk × T 7→ R1 , where T is a set of common time instances for all k systems and Yi are the sets of all functions from T into the outputs Yi . Furthermore defining a shift operator στ s.t. (στ y)(t) = y(t + τ ), we call the solutions x1 (·), . . . , xk (·) of systems Σ1 , . . . , Σk with initial conditions x1 (0), . . . , xk (0) asymptotically synchronized w.r.t the functionals g1 , . . . , gℓ if: gj (στ1 y1 (·), . . . , στk yk (·), t) ≡ 0

∀ j = 1, . . . , ℓ (3.1)

is valid for t → ∞ and some στi ∈ T. Definition 3.2: (Approximate Asymptotic Synchronization). Using the notations introduced in definition 3.1, we call systems Σ1 , . . . , Σk approximately asymptotically synchronized w.r.t. to the functionals g1 , . . . , gℓ if for some sufficiently small ε > 0: |gj (στ1 y1 (·), . . . , στk yk (·), t)| 6 ε

∀ j = 1, 2, . . . , ℓ (3.2)

is valid for t → ∞ and some στi ∈ T. In the sequel definition 3.1 and 3.2 will be used to define (approximate) synchronization. 4 Example 1: Coupled Duffing Oscillators In this section experimental results with respect to two synchronizing Duffing oscillators are presented. After introducing the dynamical system an analysis of the limiting behaviour of the system is presented. Finally, both numeric and experimental results are presented and discussed. 4.1 Problem Statement & Analysis x3 k b

κd

κd

m

Definition 4.1:((Approximate) Anti-phase Synchronization). Consider two systems Σ1 and Σ2 with initial conditions x10 and x20 and corresponding solutions x1 (x10 , t) and x2 (x20 , t). Furthermore, assume that both x1 (x10 , t) and x2 (x20 , t) are periodic in time with period T. We call the solutions of x1 (x10 , t) and x2 (x20 , t) (approximately) asymptotically synchronized in anti-phase if they are (approximately) asymptotically synchronized according to definition 3.1 or 3.2, with: g(·) = x1 (·) − ασ( T ) x2 (·),

x2

x1

Before continuing with the experimental and numerical results the systems’ limiting behaviour is analyzed. In order to do so the notion of anti-phase synchronization needs to be defined:

(4.7)

2

with α ∈ R>0 a scale factor and σ( T ) a shift operator 2 over half an oscillation period.

m

M Figure 3. Schematic representation of the set-up modeling two coupled Duffing oscillators.

Consider the system as depicted in figure 3, where:, κd (qi ) = ω02 qi + ϑqi3 m

(4.1)

where qi = xi − x3 and constants ω0 , ϑ ∈ R>0 . The system under consideration represents two undriven, undamped Duffing oscillators coupled through a third common mass. The set-up depicted in figure 2 can be adjusted to model this system by defining the actuator forces as:

Using definition 4.1 it can been shown that the dynamics of the oscillators in (4.4) - (4.6) converges to anti-phase synchronization as t → ∞ (see Lemma 4.1 below). Lemma 4.1: (Global Asymptotic Stability of the Synchronization Manifold). Consider the system of nonlinear differential equations (4.4) - (4.6). The trajectories of the oscillators Σ1 and Σ2 will converge to anti-phase synchronized dynamics, according to definition 4.1 as t → ∞ for all initial conditions. Proof (of Lemma 4.1). Consider the system (4.4) - (4.6). To analyze the limit behaviour of this system, the total energy is proposed as a candidate Lyapunov function: ξi

Fi = κi (qi ) + βi (q˙i ) − κd (qi ), i = 1, 2 (4.2)

F3 = 0

(4.3)

Where F3 = 0 is chosen because, in the original set-up, the beam already models the situation as depicted in figure 3 (linear stiffness and damping) fairly accurate. The equations of motion of the resulting system are: m¨ x1 = −κd (x1 − x3 )

(4.4)

Mx ¨3 = κd (x1 − x3 ) + κd (x2 − x3 ) − kx3 − bx˙ 3 ,

(4.6)

m¨ x2 = −κd (x2 − x3 )

(4.5)

where k, b ∈ R>0 are the stiffness and damping coefficients of the beam.

3 3 Z X 1X 2 V= mi x˙ i + κi (s) ds, 2 i=1 i=1

(4.8)

0

where m1 = m2 = m, m3 = M , ξi = xi − x3 , i = 1, 2, ξ3 = x3 , κi (qi ) = κd (qi ) and κ3 = kx3 . Calculating the time derivative of V along the solutions of the system (4.4) - (4.6) yields: V˙ = −bx˙ 23 .

(4.9)

Hence, we find V˙ ≤ 0 and the system may be analyzed using LaSalle’s invariance principle. Equation (4.9) implies that V is a bounded function of time. Moreover, xi (t) is a bounded function of time and will converge to a limit set where V˙ = 0. On this limit set x˙ 3 = x ¨3 = 0, according to (4.9). Substituting this in system (4.4) - (4.6) yields x3 = 0 on the

5

|x1 + x2 | [mm]

systems’ limit set. Substituting x3 = x˙ 3 = x¨3 = 0 in (4.6) shows:

4 3

κd (x1 ) = −κd (x2 )

(4.10)

2 1

Since κd is a one-to-one, odd function, this implies:

20

25

30

25

30

35

40

45

35

40

45

0.8

x1 = −x2

(4.11)

x˙ 2 = −x˙ 1 .

0.4 0.2 0 −0.2 −0.4

(4.12)

20

t [s]

4.2 Experimental & Numerical Results In order to experimentally investigate the synchronizing behaviour of two coupled Duffing oscillators the set-up has been modified as specified in the previous paragraph. The oscillators are released from an initial displacement of −3 mm and −2.5 mm respectively (approximately in phase) and allowed to oscillate freely. Figure 4 shows the sum of the positions of the oscillators and the position of the beam v.s. time. As becomes clear from figure 4, approximate anti-phase synchronization occurs within 40 s. Furthermore, figure 5 shows the limiting behaviour of both oscillators and the beam. Although the amplitudes of the oscillators differ significantly, the steady state phase difference is 1.01π. The most probable cause for the amplitude difference is the fact that the oscillators are not exactly identical. As a result, the beam does not come to a complete standstill, although it oscillates with an amplitude that is roughly ten times smaller than that of the oscillators. In addition to the experimental results, numerical results are provided in figure 6 and 7. The parameters in the simulation are chosen as shown in table 2. Parameters in numerical simulation.

ωo = 15.26

ϑ = 8.14

M = 0.8

m=1

k=1

b=5

The results presented in figure 6 and 7 correspond to the experimental results provided in 4 and 5 respectively. Although the oscillation frequencies of the oscillators are almost equal (within 5%) in the simulation

1

xi [mm]

The next paragraph will present numerical and experimental results that support the analysis provided in this section.

Figure 4. Experimental results: (top) Sum of the displacements of both oscillators. (bottom) Displacement of the connecting beam.

0.5

0

−0.5

−1 45

45.2

45.4

45.6

45.8

45.2

45.4

45.6

45.8

46

46.2

46.4

46.6

46.8

46

46.2

46.4

46.6

46.8

0.06 0.04

x3 [mm]

Summarizing, it has been shown that any solution of (4.4) - (4.6) will converge to anti-phase synchronized motion according to definition 4.1 as t → ∞. 

Table 2.

x3 [mm]

Finally, substituting x1 = −x2 in (4.4) - (4.5) yields:

0.6

0.02 0 −0.02 −0.04 −0.06 45

t [s] Figure 5. Experimental results: Steady state behaviour of the system. (top) Displacement of the oscillators (- x1 , - x2 ). (bottom) Displacement of the connecting beam.

and the experiment, the final amplitudes of the oscillators differs by a factor 15. This is due to the fact that in the experiment the damping is over compensated, resulting in larger amplitudes of the oscillators. This presents no problem since the residual energy may dissipate through the motion of the beam, which does not come to a complete standstill due to the amplitude difference between the oscillators. In the numerical simulation almost exact anti-phase synchronization with equal oscillator amplitudes is achieved and this mechanism fails. Finally, note that some of the differences between the experimental and simulation results may be coped with by tuning either the parameters of the numerical simulation or those of the set-up itself. The question of identifying a model can thus be reversed to tuning the parameters of the set-up rather than those of the model.

|x1 + x2 | [mm]

15

~g

θ3

10

5

θ1

0

0

5

10

15

20

25

30

35

40

45

θ2 k

k

x3 [mm]

1 0.5

2

1

ℓ

ℓ3

0

m

−0.5

k3

−1 0

5

10

15

20

25

30

35

40

ℓ m

M

45

t [s] Figure 6.

3

b

Numerical results: (top) Sum of the displacements of both Figure 8. Schematic representation of the set-up modeling two coupled rotating elements.

oscillators. (bottom) Displacement of the connecting beam.

xi [mm]

0.05

The equations of motion of the system depicted in figure 8 are:

0

−0.05 45

45.2

45.4

45.6

45.8

46

46.2

46.4

46.6

46.8

47

θ¨i = −ϑi (k (θi − θ3 ) + δi sin θi ) , i = 1, 2 (5.1) θ¨3 = ϑ3

0.05

2 X

x3 [mm]

j=1

45.2

45.4

45.6

45.8

46

46.2

46.4

46.6

46.8

47

t [s] Figure 7. Numerical results: Steady state behaviour of the system. (top) Displacement of the oscillators (- x1 , - x2 ). (bottom) Displacement of the connecting beam.

5

(5.2)


−k3 θ3 − b3 θ˙3 − δ3 sin θ3 ,

0

−0.05 45

k (θj − θ3 )

Example 2: Two Coupled Rotary Elements

Next to the synchronization of Duffing oscillators we investigated synchronization in a system of coupled rotating disc as depicted in figure 8. First the dynamics of the system will be specified in more detail and next experimental results will be presented.

5.1 Problem Statement Consider the system as depicted in figure 8. This system consists of three discs. Discs 1, 2 represent the oscillators and disc 3 is connected to both other discs by torsion springs with stiffness k. Each of the discs has an eccentric mass at a distance ℓi from it’s center (ℓ1 = ℓ2 = ℓ). Furthermore the middle disc is coupled to the world by a torsion spring with stiffness k3 and a torsion damper with constant b. The rotation of the discs is represented w.r.t. the world by the angles θi .

with ϑi = mℓ21+Ji and δi = mi gℓi . The modification i to the set-up is now more involved than in the previous example. First of all, the translation coordinates xi should be mapped to rotation angles θi (arbitrary mapping). Secondly, in case of the Duffing oscillator the actuation forces F1 and F2 were meant to act on both the oscillators and the connecting mass. In the situation depicted in figure 8 the actuation force generated to model the coupling between the oscillator discs and the middle disc by means of the torsion spring should again act on the oscillators and the connecting beam in our set-up. However, the part of the actuation force that models the influence of gravity on the oscillators should only act on the oscillators and not on the connecting beam, since in figure 8 the gravity on discs 1 and 2 exerts a force only on the corresponding disc and not directly on the middle mass. In order to adjust the set-up in figure 2 to model the system in figure 8 the actuator forces are defined as: Fi = κi (qi ) + βi (q˙i ) − ϑi (ηi + gi ) , i = 1, 2 (5.3)

F3 = κ3 (x3 ) − ϑ3 (η3 + g3 ) − g˜(·),

(5.4)

with κi (qi ) and βi (q˙i ) as defined earlier, ηi = 2 P k (θi − θ3 ) , i = 1, 2, gi = δi sin θi and g˜ = ϑi gi . j=1

Damping is left to be the natural damping of the beam

in the set-up. Furthermore, translation is mapped to rotation angles according to: θi = π2 xx⋆i , with x⋆i is the i maximal displacement of the oscillators and the beam, assuring ± 90◦ turns in the rotation space. 5.2 Experimental Results Experimental results, are presented in figure 9 and 10. It becomes clear that approximate anti-phase synchronization occurs after about 20 s. Again complete synchronization does not occur because the oscillators are not identical. In addition figure 10 shows the steady state behaviour of the rotating system, from which the approximate anti-phase synchronized behaviour becomes clear immediately.

6 Conclusion & Future Research We presented a set-up capable of conducting synchronization experiments with a variety of different oscillators. Two sets of experimental results were provided that show the potential of this set-up. First we modeled and experimentally obtained synchronization between two coupled Duffing oscillators. Second, we showed that it is possible to model systems with rotating dynamics and to effectively model the local influence of gravity in this case. In addition to studying uncontrolled synchronization the set-up has the potential to study controlled synchronization. Furthermore, we aim to model the Huygens set-up and perform controlled and uncontrolled synchronization experiments with this type of dynamical system.

7 Acknowledgements This work was partially supported by the DutchRussian program on interdisciplinary mathematics ’Dynamics and Control of Hybrid Mechanical Systems’ (NWO grant 047.017.018).

References Bennett, M., M. Schat, H. Rockwood and K. Wiesenfeld (2002). Huygens’s clocks. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 458(2019), 563–579. Blekhman, I.I., A.L. Fradkov, H. Nijmeijer and A.Yu. Pogromsky (1997). On self-synchronization and controlled synchronization. System & Control Letters (31), 299 – 305. Kuznetsov, N.V., G.A. Leonov, H. Nijmeijer and A.Yu. Pogromsky (2007). Synchronization of two metronomes,. 3rd IFAC Workshop ’Periodic Control Systems’. St. Petersburg, Russian Federation. Oud, W., H. Nijmeijer and A.Yu. Pogromsky (2006). A study of huygens synchronization. experimental results. Proceedings of the 1st IFAC Conference on Analysis and Control of Chaotic Systems, 2006. Pantaleone, J. (2002). Synchronization of metronomes. American Jounal of Physics 70(10), 992–1000. Pikovsky, A., M. Rosenblum and J. Kurths (2001). Synchronization, A universal concept in nonlinear sciences. Cambridge Nonlinear Science Series. Cambridge University Press. Pogromsky, A.Yu, V.N. Belykh and H. Nijmeijer (2003). Controlled synchronization of pendula. Proceedings 42rd IEEE Conference on Decision and Control pp. 4381–4368. Pogromsky, A.Yu, V.N. Belykh and H. Nijmeijer (2006). Group Coordination and Cooperative Control. Chap. A Study of Controlled Synchronization of Huygens’ Pendula, pp. 205–216. Vol. 336 of Lecture Notes in Control and Information Sciences. Springer. Senator, M. (2006). Synchronization of two coupled escapement-driven pendulum clocks. Journal of Sound and Vibration 291(3-5), 566–603. Tillaart, M.H.L.M. van den (2006). Design of a mechanical synchronizing system for research and demonstration purooses for d&c. Master’s thesis. Eindhoven University of Technology.

8

140

6

|θ1 + θ2 | [◦ ]

120

4

θi [◦ ]

100 80 60 40 20

2 0 −2 −4 −6

0

10

20

30

40

50

60

70

78

78.2

78.4

78.6

78.8

79

79.2

79.4

79.6

79.8

78

78.2

78.4

78.6

78.8

79

79.2

79.4

79.6

79.8

1.4 30

1.2 1

θ3 [◦ ]

θ3 [◦ ]

20 10 0

0.8 0.6 0.4

−10

0.2

−20 0

10

20

30

40

50

60

70

t [s]

t [s] Figure 9.

Experimental results: (top) Sum of the rotation angles of

the outer discs. (bottom) Rotation angle of the connecting disc.

Figure 10.

Experimental results: Steady state behaviour of the sys-

tem. (top) Outer discs (- θ1 , - θ2 ). (bottom) Connecting disc.

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