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Conservative Dynamical Systems in Oscillating Origami Tessellations
Abstract
Folded surfaces of origami tessellations sometimes exhibit non-trivial behaviors, which have attracted much attention. The oscillation of tubular waterbomb tessellation is one example. Recently, the authors reported that the kinematics of waterbomb tube depends on the discrete dynamical system that arises from the geometric constraints between modules and quasi-periodic solutions of the dynamical system generate oscillating configurations. Although the quasi-periodic behavior is the characteristic of conservative systems, whether the system is conservative has been unknown. In this paper, we decompose the dynamical system of waterbomb tube into three steps and represent the one-step using the two kinds of mappings between zigzag polygonal linkages. By changing parameters of the mappings and composite them, we generalize the dynamical system of waterbomb tube to that of various tubular origami tessellations and show their oscillating configurations. Furthermore, by analyzing the mapping, we give proof of the conservation of the dynamical system.
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Significance
The traditional approach to designing origami metamaterials uses particular, highly symmetric crease patterns to generate folding motions for reconfigurability. We instead consider origami sheets with periodic but otherwise generic, asymmetric triangular faces and show they exhibit nonlinear folding motions which transform sheets through two-dimensional families of cylindrical configurations, with the addition of quadrilateral faces restricting sheets to one-dimensional subsets of configurations. This leads to a topological class of mechanical modes, preventing origami from exhibiting exponentially localized floppy modes observed in other systems. These results do not depend on scale or material and hence have applications extending to architecture and robotics, but particularly to the nanoscale, where limited control over fold patterns can constrain traditional techniques.
Origami has recently emerged as a promising building block of mechanical metamaterials because it offers a purely geometric design approach independent of scale and constituent material. The folding mechanics of origami-inspired metamaterials, i.e., whether the deformation involves only rotation of crease lines (rigid origami) or both crease rotation and facet distortion (nonrigid origami), is critical for fine-tuning their mechanical properties yet very difficult to determine for origami patterns with complex behaviors. Here, we characterize the folding of tubular waterbomb using a combined kinematic and structural analysis. We for the first time uncover that a waterbomb tube can undergo a mixed mode involving both rigid origami motion and nonrigid structural deformation, and the transition between them can lead to a substantial change in the stiffness. Furthermore, we derive theoretically the range of geometric parameters for the transition to occur, which paves the road to program the mechanical properties of the waterbomb pattern. We expect that such analysis and design approach will be applicable to more general origami patterns to create innovative programmable metamaterials, serving for a wide range of applications including aerospace systems, soft robotics, morphing structures, and medical devices.
This dissertation conducts kinematic analysis of spatial linkages ranging from spherical linkages to overconstrained linkages based on the D-H matrix method, and applies it to explore the rigid foldability and motion behaviour of origami patterns. In this process, the rigid foldability of triangle twist origami pattern is firstly examined based on the kinematics of spherical 4 R linkage network and new overconstrained 6 R linkages are derived by kirigami technique. Then the kinematics of the plane-symmetric Bricard 6 R linkage is analyzed and its bifurcation variations are discussed. After that, the results are applied to study the symmetric folding of six-crease thick-panel waterbomb origami, which is modelled as a network of planesymmetric Bricard 6 R linkages. The motion behaviour of its corresponding tessellation of zerothickness sheet is demonstrated by a network of spherical 6 R linkages. Finally, the motion behaviour of the closed cylindrical form of waterbomb origami is investigated through a parametric study, by means of modelling it as a closed network of spherical 6 R linkages. These studies help to deepen the understanding of spatial linkage kinematics and rigid origami motion, and lay the foundation for engineering applications of spatial linkages and rigid origami patterns.
Transforming flat sheets into three-dimensional structures has emerged as an exciting manufacturing
paradigm on a broad range of length scales. Among other advantages, this technique permits the use of
functionality-inducing planar processes on flat starting materials, which after shape-shifting, result in
a unique combination of macro-scale geometry and surface topography. Fabricating arbitrarily
complex three-dimensional geometries requires the ability to change the intrinsic curvature of initially
flat structures, while simultaneously limiting material distortion to not disturb the surface features.
The centuries-old art forms of origami and kirigami could offer elegant solutions, involving only
folding and cutting to transform flat papers into complex geometries. Although such techniques are
limited by an inherent developability constraint, the rational design of the crease and cut patterns
enables the shape-shifting of (nearly) inextensible sheets into geometries with apparent intrinsic
curvature. Here, we review recent origami and kirigami techniques that can be used for this purpose,
discuss their underlying mechanisms, and create physical models to demonstrate and compare their
feasibility. Moreover, we highlight practical aspects that are relevant in the development of advanced
materials with these techniques. Finally, we provide an outlook on future applications that could
benefit from origami and kirigami to create intrinsically curved surfaces.
Origami tessellations are particular textured morphing shell structures. Their unique folding and unfolding mechanisms on a local scale aggregate and bring on large changes in shape, curvature and elongation on a global scale. The existence of these global deformation modes allows for origami tessellations to fit non-Trivial surfaces thus inspiring applications across a wide range of domains including structural engineering, architectural design and aerospace engineering. The present paper suggests a homogenization-Type two-scale asymptotic method which, combined with standard tools from differential geometry of surfaces, yields a macroscopic continuous characterization of the global deformation modes of origami tessellations and other similar periodic pin-jointed trusses. The outcome of the method is a set of nonlinear differential equations governing the parametrization, metric and curvature of surfaces that the initially discrete structure can fit. The theory is presented through a case study of a fairly generic example: The eggbox pattern. The proposed continuous model predicts correctly the existence of various fittings that are subsequently constructed and illustrated. © 2017 The Author(s) Published by the Royal Society. All rights reserved.
Folded surfaces of origami tessellations have attracted much attention because they sometimes exhibit non-trivial behaviors. It is known that cylindrical folded surfaces of waterbomb tessellation called waterbomb tube can transform into wave-like surfaces, which is a unique phenomenon not observed on other tessellations. However, the theoretical reason why wave-like surfaces arise has been unclear. In this paper, we provide a kinematic model of waterbomb tube by parameterizing the geometry of a module of waterbomb tessellation and derive a recurrence relation between the modules. Through the visualization of the configurations of waterbomb tubes under the proposed kinematic model, we classify solutions into three classes: cylinder solution, wave-like solution, and finite solution. Furthermore, we give proof of the existence of a wave-like solution around one of the cylinder solutions by applying the knowledge of the discrete dynamical system to the recurrence relation.
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