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Sphere Assemblies: Control of Rotation States and Construction of Space-Filling Packings

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Sphere Assemblies: Control of Rotation States and Construction of Space-Filling Packings Research Collection Doctoral Thesis Sphere assemblies: control of rotation states and construction of space-filling packings Author(s): Stäger, Dominik V. Publication Date: 2016 Permanent Link: https://doi.org/10.3929/ethz-a-010780463 Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library Diss. ETH No. 23864 Sphere Assemblies: Control of Rotation States and Construction of Space-Filling Packings A thesis submitted to attain the degree of Doctor of Sciences of ETH Zurich (Dr. sc. ETH Zurich) presented by Dominik Valerian Stager¨ MSc. Materials Science, ETH Zurich born 18.06.1989 citizen of Glarus S¨ud GL, Switzerland accepted on the recommendation of Prof. Dr. Hans J. Herrmann, examiner Prof. Dr. Tomaso Aste, co-examiner Prof. Dr. Ronald Peikert, co-examiner 2016 Acknowledgment I would like to thank Prof. Dr. Hans J. Herrmann for proposing me the topic for my PhD, which fascinated me from the beginning to the end, and which led to fruitful results. Furthermore, I would like to thank him for the supervision of this thesis, especially for the numerous constructive criticism. Special thanks goes to Prof. Dr. Nuno A. M. Ara´ujo for his supervision through a large period of my PhD. I highly appreciated the support of his sharp and incredibly fast mind. I would like to thank Dr. Falk Wittel for his support regarding the con- struction of experiments, and Sergio Solorzano Rocha for fruitful discussions about my research. I am grateful to Prof. Dr. Tomaso Aste and Prof. Dr. Ronald Peikert for accepting the request to co-examine this thesis and for showing interest in my work. Special thanks goes to my coworkers for the amazing company throughout my PhD. After all this time, I feel like being part of a big family with a lot of awesome brothers. To see so many friends at work every single day is an enormous motivation and an incredible source of energy. I do not want to have missed out on any of the great lunch breaks, and neither any of the numerous laughs we shared every day. I am very grateful for all the good moments that we could share apart from work and the unforgettable stories that resulted from them. Last but not least, I want to thank my family and all my friends with whom I could share plenty of good times that let me enjoy life to its fullest during this special period. Contents Kurzfassung i Abstract iii Related Publications v 1 Introduction 1 1.1 Overview ............................. 1 1.2 Background ........................... 3 1.3 Motivation ............................ 5 1.3.1 Understanding of Rotational Dynamics ........ 5 1.3.2 Search for Further 3D Space-Filling Bearings ..... 6 2 Prediction and Control of Slip-Free Rotation States in Sphere Assemblies 7 2.1 Introduction ........................... 7 2.2 Kinetics of Bipartite Assemblies ................ 9 2.3 Construction of 4DOF Assemblies ............... 11 2.4 Prediction of the Final Slip-Free State ............ 16 2.5 Control of the Slip-Free State ................. 19 2.5.1 Control Within the Model ............... 20 2.5.2 Control in Experiment ................. 22 2.6 Experimental Details ...................... 26 2.7 Final Remarks .......................... 27 3 Construction of Self-Similar Space-Filling Sphere Packings in Three and Four Dimensions 29 3.1 Introduction ........................... 29 3.2 Basic Idea of Generating a Packing .............. 31 3.3 Circle Inversion ......................... 31 3.3.1 Basic Properties ..................... 31 3.3.2 Multiple Inversion Circles ............... 33 3.3.3 Mathematics of Circle Inversion ............ 35 3.4 Constraints on the Generating Setup ............. 37 3.5 How to Construct Generating Setups ............. 41 3.5.1 Construction of 2D Generating Setups ........ 41 3.5.2 Generalization to Higher Dimensions ......... 42 3.5.3 Determine Positions and Radii of Setup Elements .. 47 3.6 Discovered Packings ...................... 53 3.6.1 Fractal Dimension ................... 53 3.6.2 Contact Network .................... 55 3.6.3 Topological Comparison ................ 67 3.7 Modified Packings ........................ 71 3.8 Final Remarks .......................... 71 4 Cutting Self-Similar Space-Filling Sphere Packings 75 4.1 Introduction ........................... 75 4.2 Properties of Packings ..................... 77 4.3 Cutting ............................. 77 4.3.1 Random Cuts ...................... 79 4.3.2 Special Cuts ....................... 84 4.4 Final Remarks .......................... 92 5 Conclusion 95 References 97 Kurzfassung Wir widmen uns zwei verschiedenen Themenbereichen. Wir untersuchen die Rotationsdynamik von aneinanderliegenden, individuell rotierenden Kugeln, welche sich gegenseitig beeinflussen. Daruber¨ hinaus entwickeln wir neue Konstruktionsmethoden fur¨ raumfullenden¨ Kugelpackungen. Zuerst studieren wir die Rotationsdynamik von aneinanderliegenden Ku- geln, sogenannten Kugelclustern. Dabei nehmen wir an, dass die Kugeln an ihrer Position fixiert sind, jedoch frei rotieren k¨onnen. Weiter befassen wir uns nur mit bipartiten Kugelclustern. Ein Kugelcluster ist bipartit, falls man es mit nur zwei Farben so einf¨arben kann, sodass sich keine Kugeln gleicher Farbe beruhren.¨ Jedes bipartite Kugelcluster hat schlupffreie Rota- tionszust¨ande, in welchen sich alle Kugeln gleichzeitig drehen k¨onnen, ohne Schlupf zwischen beruhrenden¨ Kugeln. Mit einem Modell, welches lediglich Gleitreibung berucksichtigt,¨ untersuchen wir, wie sich bipartite Kugelclus- ter von einem Anfangszustand mit frei w¨ahlbaren Winkelgeschwindigkeiten zu einem schlupffreien Endzustand bewegen. Dabei stellen wir fest, dass ge- wisse Summen von Variablen, welche die Massen, Radien, Winkelgeschwin- digkeiten und Positionen der Kugeln einbeziehen, zeitlich unver¨anderlich sind. Das bedeutet, dass diese vom Anfangszustand bis hin zum schlupf- freien Endzustand konstant bleiben. Fur¨ gewisse bipartite Kugelcluster ist der schlupffreie Zustand eindeutig bestimmt fur¨ gegebene Werte der zeit- lich konstanten Summen. In diesem Fall kann man den Endzustand fur¨ jeden beliebigen Anfangszustand exakt vorhersagen. Diese vorhersagbaren Kugelcluster mussen¨ genau vier Freiheitsgrade im schlupffreien Zustand ha- ben. Glucklicherweise¨ sind solche einfach zu konstruieren und k¨onnen aus i nur zwei, aber auch aus viel mehr Kugeln bestehen, theoretisch aus unbe- grenzt vielen. Uberraschenderweise¨ ist der schlupffreie Endzustand solcher Kugelcluster unabh¨angig von der St¨arke der Gleitreibung zwischen den Ku- geln. Desweiteren l¨asst sich der schlupffreie Zustand kontrollieren. Durch die externe Kontrolle von lediglich zwei beliebigen Kugeln kann jeder m¨ogliche schlupffreie Rotationszustand des Kugelclusters kontrolliert werden, was wir auch experimentell demonstrieren. Im schlupffreien Zustand eines jeden bi- partiten zweidimensionalen Clusters gleich grosser Scheiben sind alle Rota- tionsgeschwindigkeiten identisch. Nicht so in einem dreidimensionalen Ku- gelcluster, wo Kugeln gleicher Grsse unterschiedliche Rotationsgeschwindig- keiten haben knnen. Deshalb ist es m¨oglich in einem kontrollierbaren Ku- gelcluster die Kugeln entlang einer frei w¨ahlbaren Richtung zu beschleuni- gen, was eine zuvor unbekannte mechanische Funktionalit¨at darstellt. Desweiteren befassen wir uns mit raumfullenden¨ Kugelpackungen. Im Detail behandeln wir nur Packungen, welche mittels Kugelinversionen erzeugt wer- den k¨onnen, was immer exakt selbst¨ahnliche Packungen sind. Diese Packun- gen sind fraktal und die Gr¨ossenverteilung ihrer Kugeln folgt asymptotisch einem Potenzgesetz, woraus man die fraktale Dimension absch¨atzen kann. Inspiriert durch vorhergehende Arbeiten entwickeln wir eine Konstruktions- methode in zwei Dimensionen, welche wir fur¨ jede h¨ohere Dimension verall- gemeinern. Mittels der neuen Konstruktionsmethode finden wir zahlreiche neue Topologien in drei und vier Dimensionen. Zus¨atzlich stellen wir eine Strategie vor um neue niedrigerdimensionale Topologien aus bereits ent- deckten zu schneiden. Im Ganzen erm¨oglicht uns dies weitere raumfullende¨ Topologien in beliebigen Dimensionen zu finden. Die zahlreichen Topologien die wir finden weisen eine grosse Bandbreite an fraktalen Dimensionen auf und k¨onnen als eine Auswahl fur¨ ideal dichte Packungen mit verschiedenen Gr¨ossenverteilungen gesehen werden. ii Abstract We study two different areas. We investigate the rotational dynamics of assemblies of touching, individually rotating spheres, which influence each other’s rotation. Beyond that, we develop new methods to construct space- filling sphere packings. First, we study the rotational dynamics of an assembly of contacting spheres. We are interested in the case where the spheres can not move in space, but are allowed to rotate. We only deal with bipartite assemblies, i.e., assemblies where one can color the spheres using only two colors such that no spheres of same color touch. Any bipartite assembly of spheres has slip-free rotation states, i.e., it is possible that all spheres rotate without any slip between contacting spheres. With a model that only considers sliding friction, we investigate how bipartite assemblies drive from an initial rotation state with arbitrary angular velocities toward a slip-free state. We find that certain sums of variables are time-invariant, i.e., they stay constant during the dy- namics toward the slip-free state. These sums involve the individual
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