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Curved Folding and Developable Surfaces in Mechanism and Deployable Structure Design Todd G Brigham Young University BYU ScholarsArchive All Theses and Dissertations 2018-06-01 Art to Engineering: Curved Folding and Developable Surfaces in Mechanism and Deployable Structure Design Todd G. Nelson Brigham Young University Follow this and additional works at: https://scholarsarchive.byu.edu/etd Part of the Mechanical Engineering Commons BYU ScholarsArchive Citation Nelson, Todd G., "Art to Engineering: Curved Folding and Developable Surfaces in Mechanism and Deployable Structure Design" (2018). All Theses and Dissertations. 6865. https://scholarsarchive.byu.edu/etd/6865 This Dissertation is brought to you for free and open access by BYU ScholarsArchive. It has been accepted for inclusion in All Theses and Dissertations by an authorized administrator of BYU ScholarsArchive. For more information, please contact [email protected], [email protected]. Art to Engineering: Curved Folding and Developable Surfaces in Mechanism and Deployable Structure Design Todd G Nelson A dissertation submitted to the faculty of Brigham Young University in partial fulfillment of the requirements for the degree of Doctor of Philosophy Larry L. Howell, Chair Spencer P. Magleby Anton E. Bowden David T. Fullwood David W. Jensen Robert J. Lang Department of Mechanical Engineering Brigham Young University Copyright © 2018 Todd G Nelson All Rights Reserved ABSTRACT Art to Engineering: Curved Folding and Developable Surfaces in Mechanism and Deployable Structure Design Todd G Nelson Department of Mechanical Engineering, BYU Doctor of Philosophy This work investigates how curved-crease origami and the developable surfaces which compose it can be transitioned to engineering design. Methods for creating flexible, tailorable- property surfaces that function as thick panels in place of paper are presented. Concepts from curved-crease origami and developable surfaces that can describe and extend engineering appli- cations are discussed and demonstrated. These concepts are particularly beneficial to applications where curved surfaces are integral to the function, deployability is desired, and planar manufactur- ing could be beneficial. The first part of this work uses arrays of compliant elements to create flexible-tailorable property surfaces. The key feature to these arrays is the alignment of the most flexible bending axis of the individual elements to the ruling line arrangement of a developable surface. This alignment can enable bending of thick panels while maintaining lower stresses, a quality necessary for the transitioning of curved-crease origami into thick materials. The stiffness and stress of these arrays is modeled and physical prototypes are demonstrated. Additionally, shape factors are developed for these compliant arrays (CAs) to facilitate material selection for the panels and understand how the geometry of the array changes the effective properties of the panel. The second part of this work describes and demonstrates several concepts of curved-crease origami and developable surfaces that can benefit mechanism and structure design, particularly in the context of rolling-contact mechanisms. The design of a rolling-contact joint connected by flexible bands similar to a Jacob’s Ladder toy is extended through incorporating curved creases into the design. The resulting design is deployable from a compact state to a functional state and can be manufactured from a single plane and folded into shape. Mathematical formulations are presented to describe the classes of developable surfaces in terms of properties which are frequently important in mechanism design. These natural equa- tions for a single class of developable surface are conducive to modeling the folding motion of rigid-ruling developables, developables whose ruling lines do change location in a surface during folding. These formulations are used to generalize the design of rolling-contact joints to a family of joints capable of single degree of freedom spatial motions, being manufactured from a plane, and exhibiting a tailorable force response. Finally practical design suggestions for the implementa- tion of rolling-contact joints is given. These include methodology to create sunken flexures which serve to increase the normal force between rolling bodies to prevent slip. Keywords: developable, curved folding, origami, deployable ACKNOWLEDGMENTS I feel an appropriate start to an acknowledgements section would be with the often quoted words of Issac Newton, “If I have seen further it is by standing on the shoulders of giants.” Not only have the academic works that this research is built upon been invaluable, but also and perhaps more importantly has been the influence of those I have interacted with face to face while compiling this work. I would like to thank my PhD committee, Spencer Magleby, Anton Bowden, David Full- wood, David Jensen, Robert Lang, and particularly my advisor Larry Howell. Without their kind- ness, direction, and willingness to share their experience this work would not have come to pass. I would also like to acknowledge the gracious advising I received from Just Herder during my time in Delft. My peers in the Compliant Mechanisms Research Group have not only provided construc- tive insights and discussions, but have created an atmosphere where a strong work ethic, high moral integrity, and positivity had prominent places. It has been a pleasure to associate with them nearly daily. Particularly I would like to thank Jared Bruton, Jason Dearden, Jason Lund, Patrick Walton, Nathan Pehrson, Thomas Evans, Nathan Rieske, Alex Avila, Jared Butler, Jessica Morgan, Jacob Badger, Ezekiel Merriam, Kyler Tolman, John Sessions, Trent Zimmerman, Nichole Cross, Bryce Barker, Alden Yellowhorse, Janette Herron, Peter Schleede, Issac Delimont, and Eric Wilson for the various insights and ideas they have brought to this work. I also appreciated working with the visiting scholars in the group Zhicheng Deng, Liping Zhang, and Guimin Chen. While I was only able to spend a short time with them, I would like to thank my colleagues in the Precision and Microsystems Engineering group at TU Delft for welcoming me there and for the insightful discussions. The staff and other faculty at Brigham Young University have been phenomenal, with spe- cial mention of Kevin Cole, Terri Bateman, Jeffrey Niven, and David Morgan. I regret that I cannot list all the people personally who have impacted this work whether directly or indirectly, though to some it may seem like I tried. If you feel like you were left out, reach out to me and I’d be happy to thank you personally. I would like to acknowledge the funding sources that made this work possible. They are the National Science Foundation and the Air Force Office of Scientific Research under NSF Grant EFRI-ODISSEI-1240417 and the National Science Foundation Graduate Research Fellowship Pro- gram under Grant No. 1247046. I was also fortunate to participate in the NSF Graduate Research Opportunities Worldwide (GROW) program as part of the National Science Foundation Graduate Research Fellowship Program (NSF GRFP). I am grateful to my parents for their continual support and patience with my interest in engineering and taking things apart. My wife Kambria and daughter Sofie motivate me to be my best self and I am grateful for their love and encouragement. Most of all I thank my Father in Heaven and His son Jesus Christ. The peace, purpose, and inspiration they provide is not found anywhere else. TABLE OF CONTENTS LIST OF TABLES ....................................... viii LIST OF FIGURES ...................................... ix Chapter 1 Introduction ................................... 1 1.1 Problem Statement . 1 1.2 Background . 1 1.3 Research Hypotheses . 5 1.3.1 Hypothesis 1 . 5 1.3.2 Hypothesis 2 . 6 1.4 Overview of Organization . 8 Chapter 2 Facilitating Deployable Mechanisms and Structures via Developable Lam- ina Emergent Arrays .............................. 9 2.1 Introduction . 9 2.2 Background . 10 2.2.1 Developable Surfaces . 10 2.2.2 Lamina Emergent Mechanisms . 12 2.2.3 Current and Past Applications . 17 2.3 Method . 18 2.4 Mathematical Model . 21 2.5 Conical Array Pattern Generation . 24 2.6 Structure and Mechanism Examples . 26 2.6.1 Subtractive Manufacturing . 27 2.6.2 Additive Manufacturing (3D printing) . 31 2.7 Conclusion . 33 Chapter 3 Material Selection Shape Factors for Compliant Arrays in Bending .... 35 3.1 Introduction . 35 3.2 Background . 36 3.2.1 Compliance . 36 3.2.2 CA Applications: Deployable Structures and Developable Surfaces . 37 3.2.3 Shape Factors and Material Selection . 38 3.3 Methods . 39 3.3.1 Compliant Array Geometry . 39 3.3.2 Analytical Characterization of Bending Shape Factors and Strength Effi- ciency Factors . 45 3.3.3 Experimental Verification . 51 3.4 Results and Discussion . 54 3.4.1 CA Shape Factor Analysis . 54 3.4.2 Material Selection with CAs . 57 3.5 Conclusions . 61 v Chapter 4 Curved-Folding-Inspired Deployable Compliant Rolling-contact Element (D-CORE) .................................... 64 4.1 Introduction . 64 4.2 Background . 65 4.2.1 Jacob’s Ladder . 66 4.2.2 Compliant Mechanisms . 66 4.2.3 Contact-Aided Rolling Joints . 67 4.2.4 Origami Inspired Mechanisms . 68 4.3 Deployable CORE (D-CORE) . 69 4.3.1 Description . 69 4.3.2 Physical Models . 79 4.4 Deployable Translating Platform . 82 4.4.1 Description . 82 4.4.2 Physical Model . 84 4.5 Discussion . 84 4.6 Conclusion . 85 Chapter 5 Natural Equations for Rigid-Ruling Single-Class Developable Surfaces .. 86 5.1 Introduction . 87 5.2 Background . 88 5.2.1 Natural Equations of Curves . 88 5.2.2 Intrinsic properties of surfaces . 89 5.2.3 Normal, Principal, Geodesic, and Gaussian Curvatures . 89 5.2.4 Darboux Frame (T, L, U frame) . 90 5.2.5 Developable Surfaces . 91 5.3 Natural Equations of Rigid-Ruling Single-Class Developable Surfaces . 94 5.3.1 Cross Ruling Principal Curves (Cross Curves) . 95 5.3.2 Canonical Developable Parametrizations . 97 5.3.3 Ruling Curvature . 99 5.3.4 Osculating Shapes . 100 5.3.5 Ruling Curvature Related to Principal Curvature Variation . 101 5.3.6 Developable Surface Formulas .
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