The research results presented in this thesis provide tools and methods to aid in the design of developable mechanisms. This work will help engineers design compact mechanisms onto developable surfaces, making it possible for them to be used in future applications. The thesis introduces terminology and definitions to describe conical developable mechanisms. Models are developed to describe mechanism motion with respect to the apex of the conical surface, and connections are made to cylindrical developable mechanisms using projected angles. The Loop Sum Method is presented as an approach to determine the geometry of the cone to which a given spherical mechanism can be mapped. A method for position analysis is presented to determine the location of any point along the link of a mechanism with respect to the conical geometry. These methods are also applied to multiloop spherical mechanisms. This work created tools and methods to design cylindrical and conical developable mechanisms from flat, planar patterns. Equations are presented that relate the link lengths and link angles of planar and spherical mechanisms to the dimensions in a flat configuration. These flat patterns can then be formed into curved, developable mechanisms. Guidelines are established to determine if a mechanism described by a flat pattern can exhibit intramobile or extramobile behavior. A developable mechanism can only potentially exhibit intramobile or extramobile behavior if none of the links extend beyond half of the flat pattern. The behavior of a mechanism can change depending on the location of the cut of the flat pattern. Different joint designs are discussed including lamina emergent torsional (LET) joints. It is shown that developable mechanisms on regular cylindrical surfaces can be described using cyclic quadrilaterals. Mechanisms can exist in either an open or crossed configuration, and these configurations correspond to convex and crossed cyclic quadrilaterals. Using equations developed for both convex and crossed cyclic quadrilaterals, the geometry of the reference surface to which a four-bar mechanism can be mapped is found. Grashof mechanisms can be mapped to two surfaces in open or crossed configurations. The way to map a non-Grashof mechanism to a cylindrical surface is in its open configuration. Extramobile and intramobile behavior can be achieved depending on selected pairs within a cyclic quadrilateral and its position within the circumcircle. Selecting different sets of links as the ground link changes the potential behavior of the mechanism. Different cases are tabulated to represent all possibilities.

This thesis provides resources that enable the design of novel and compact mechanical devices by providing terminology, engineering models, and design methods in the fields of developable mechanisms and origami-based engineering.The first part of this work presents engineering models to aid in the design of cylindrical developable mechanisms. These models take into account the added spatial restrictions imposed by the developable surface. Equations are provided for the kinematic analysis of cylindrical developable mechanisms. A new classification for developable mechanisms is also presented (intramobile, extramobile, and transmobile) and two graphical methods are provided for determining this classification for single-DOF planar cylindrical developable mechanisms. Characteristics specific to four-bar cylindrical developable mechanisms are also discussed. The second part addresses a key challenge in origami design: how to achieve stability while maintaining the desired folding motion. The origami stability integration method (OSIM) provides an approach for graphically combining various techniques to achieve stability. This thesis presents improvements and additions to the OSIM that allow it to be applied to many different scenarios. Existing stability techniques are also categorized into four groups based on whether they are intrinsic or extrinsic to the origami pattern and whether they exhibit gradual or non-gradual energy storage behaviors. These categorizations can help designers select appropriate techniques for their application. Four case studies are presented which use the OSIM and the technique categorization to conceptualize stability in origami-based devices.

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 applications are discussed and demonstrated. These concepts are particularly beneficial to applications where curved surfaces are integral to the function, deployability is desired, and planar manufacturing 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 equations 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 implementation 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.

In architectural and sculptural practice, the eventual goal is constructing the shapes that have been designed. Due to fabrication considerations, shapes with planar faces are in demand for these practices. In this thesis, a novel computational modeling approach to design constructible shapes is introduced. This method guarantees that the resulting shapes are planar meshes with 3-valence vertices, which can always be physically constructed using planar or developable materials such as glass, sheet metal or plywood. The method introduced is inspired by the traditional sculpture and is based on the idea of carving a mesh by using slicing planes. The process of determining the slicing planes can either be interactive or automated. A framework is developed which allows user to sculpt shapes by using the in- teractive and automated processes. The framework allows user to cut a source mesh based on its edges, faces or vertices. The user can sculpt various kinds of developable surfaces by cutting the parallel edges of the mesh. The user can also introduce in- teresting conical patterns by cutting dierent vertex, edge, face combinations of the mesh.

In architectural and sculptural practice, the eventual goal is constructing the shapes that have been designed. Due to fabrication considerations, shapes with planar faces are in demand for these practices. In this thesis, a novel computational modeling approach to design constructible shapes is introduced. This method guarantees that the resulting shapes are planar meshes with 3-valence vertices, which can always be physically constructed using planar or developable materials such as glass, sheet metal or plywood. The method introduced is inspired by the traditional sculpture and is based on the idea of carving a mesh by using slicing planes. The process of determining the slicing planes can either be interactive or automated. A framework is developed which allows user to sculpt shapes by using the in- teractive and automated processes. The framework allows user to cut a source mesh based on its edges, faces or vertices. The user can sculpt various kinds of developable surfaces by cutting the parallel edges of the mesh. The user can also introduce in- teresting conical patterns by cutting dierent vertex, edge, face combinations of the mesh.

There is an increasing need for compact mechanical systems that can accomplish sophisticated tasks. Technologies like ortho-planar and lamina emergent mechanisms (LEMs) have been developed to satisfy needs like these by stowing in planar sheets from which they emerge to perform their function. They can be compact, lightweight, monolithic, scalable, and can withstand harsh environments. They are limited, however, by their base element---planar surfaces. Applications requiring these advantages often include curved surfaces, like aircraft wings, needles, and automotive bodies. In this research, developable mechanisms are presented as a solution to satisfy the need for mechanisms that can conform to or emerge from curved surfaces. Foundational principles which enable designers to leverage the advantages of developable mechanisms are described herein.Developable mechanisms result from the union of mechanisms and developable surfaces. Developable (flattenable) surfaces act as a fitting medium because of their particular advantages in manufacturability and how well they accompany four-link, revolute joint (4R) mechanisms. The definition, including specific qualifying criteria, for developable mechanisms is given. Certain types of mechanisms and classes of developable surfaces can be combined to satisy that criteria. Developable mechanism sub-classes are defined as planar, cylindrical, conical and tangent developable mechanisms. It is shown that planar and spherical mechanisms can be used to create cylindrical and conical developable mechanisms, respectively. The Bennett and other 7R mechanisms can be used for tangent developable mechanisms. Steps for developable mechanism creation are presented, and several physical prototypes are provided to demonstrate feasibility.The cylindrically curved Lamina Emergent Torsional (LET) joint is offered as an enabling technology for producing compliant developable mechanisms. A mathematical model predicting force-deflection and stress behavior is provided and verified. The relationship between stiffness and strain energy storage for curved sheet materials with incorporated LET joints is explored. Material shape factors are used to derive an effective modulus of elasticity and an effective modulus of resilience, which are compared with original values on an Ashby plot. While there is a decrease in the modulus of resilience, there is a much more significant decrease in the modulus of elasticity. A material performance index is provided as an example for determining suitable materials for a given stiffness-reduction application. It is shown that the cylindrically curved LET joint makes it possible to create highly flexible joints that maintain much of their energy storage capability in curved sheet materials.

Origami has been extensively studied by engineers for its unique motions and ability to collapse to small volumes. Techniques have been studied for replicating origami-like folding motion in thick materials, but limited practical applications of these techniques have been demonstrated. Developable mechanisms are a new mechanism type that has a similar ability to collapse to a low profile. The cylindrical developable mechanism has the ability to emerge from and conform to a cylindrical surface. In this work, a few practical applications of devices with novel expanding motions are presented. The design and testing of an origami-inspired deployable ballistic barrier, which was designed by combining and modifying existing thickness accommodation techniques, is discussed. The properties of cylindrical developable mechanisms are examined and two devices designed for use with minimally invasive surgical tooling are presented. Various hinge options for small-scale cylindrical developable mechanisms are then reviewed and discussed. A planar modeling assumption for curved lamina emergent torsional joints in thin-walled cylinders is then analytically and empirically validated. Conclusions are drawn and recommendations for future work are given.

This work develops models and tools to help designers address the challenges associated with designing origami-based and developable mechanisms. These models utilize strain energy, kinematics, compliant mechanisms, and graphical techniques to make the design of origami-based and developable mechanisms approachable and intuitive. Origami-based design tools are expanded through two methods. First presented is a generalized approach for identifying single-output mechanical advantage for a multiple-input compliant mechanism, such as many origami-based mechanisms. The model is used to predict the force-deflection behavior of an origami-based mechanism (Oriceps) and is verified with experimental data from magnetic actuation of the mechanism. Second is a folding technique for thick-origami, called the regional-sandwiching of compliant sheets (ReCS), which creates flat-foldable, rigid-foldable, and self-deploying thick origami-based mechanisms. The technique is used to create mountain/valley assignments for each fold about a vertex, constraining motion to a single branch of folding. Strain energy in deflected flexible members is used to enable self-deployment. Three physical models, a simple single-fold mechanism, a degree-four vertex mechanism, and a full tessellation, are presented to demonstrate the ReCS technique. Developable mechanism design is further enabled through an exploration of their feasible design space. Terminology is introduced to define the motion of developable mechanisms while interior and exterior to a developable surface. The limits of this motion are identified using defined conditions. It is shown that the more difficult of these conditions may be treated as a non-factor during the design of cylindrical developable mechanisms given certain assumptions. These limits are then applied to create a resource for designing bistable developable mechanisms (BDMs) that reach their second stable positions while exterior or interior to a cylindrical surface. A novel graphical method for identifying stable positions of linkages using a single dominant torsional spring, called the Principle of Reflection, is introduced and implemented. The results are compared with a numerical simulation of 30,000+ mechanisms to identify possible incongruencies. Two tables summarize the results as the guide for designing extramobile and intramobile BDMs. In fulfilling the research objectives, this dissertation contributes to the scientific community of origami-based and developable mechanism design approaches. As a result of this work, practitioners will be better able to approach and design complex origami-based and developable mechanisms.

Estudaremos as singularidades genéricas de superfécies regradas em R3. O objetivo do trabalho é mostrar que as singularidades genéricas que ocorrem no conjunto das superfícies regradas são as mesmas que ocorrem no conjunto das aplicações diferenciáveis de R2 em R3, enquanto que as singularidades genéricas das superfícies desenvolvíveis, que formam um subconjunto das superfícies regradas, são mais degeneradas. / We study generic singularities of ruled surfaces in R3. In this work we show that generic singularities appearing in the set of ruled surfaces are the same that occur in the set of map germs from R2 to R3, while the generic singularities of developable surfaces are more degenerate.

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