×

Design of rigid-foldable doubly curved origami tessellations based on trapezoidal crease patterns. (English) Zbl 1404.52026

Summary: This paper presents a mathematical framework for the design of rigid-foldable doubly curved origami tessellations based on trapezoidal crease patterns that can simultaneously fit two target surfaces with rotational symmetry about a common axis. The geometric parameters of the crease pattern and the folding angles of the target folded state are determined through a set of combined geometric and constraint equations. An algorithm to simulate the folding motion of the designed crease pattern is provided. Furthermore, the conditions and procedures to design folded ring structures that are both developable and flat-foldable and stacked folded structures consisting of two layers that can fold independently or compatibly are discussed. The proposed framework has potential applications in designing engineering doubly curved structures such as deployable domes and folded cores for doubly curved sandwich structures on the aircraft.

MSC:

52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
65D17 Computer-aided design (modeling of curves and surfaces)
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Zirbel SA, Lang RJ, Thomson MW, Sigel DA, Walkemeyer PE, Trease BP, Magleby SP, Howell LL. (2013) Accommodating thickness in origami-based deployable arrays. J. Mech. Des. 135, 111005. (doi:10.1115/1.4025372) · doi:10.1115/1.4025372
[2] Kuribayashi K, Tsuchiya K, You Z, Tomus D, Umemoto M, Ito T, Sasaki M. (2006) Self-deployable origami stent grafts as a biomedical application of Ni-rich TiNi shape memory alloy foil. Mat. Sci. Eng. A 419, 131-137. (doi:10.1016/j.msea.2005.12.016) · doi:10.1016/j.msea.2005.12.016
[3] Song Z(2014) Origami lithium-ion batteries. Nat. Commun. 5, 3140. (doi:10.1038/ncomms4140) · doi:10.1038/ncomms4140
[4] Tang R, Huang H, Tu H, Liang H, Liang M, Song Z, Xu Y, Jiang H, Yu H. (2014) Origami-enabled deformable silicon solar cells. Appl. Phys. Lett. 104, 083501. (doi:10.1063/1.4866145) · doi:10.1063/1.4866145
[5] Zhang Y, Huang Y, Rogers JA. (2015) Mechanics of stretchable batteries and supercapacitors. Curr. Opin. Solid State Mater. Sci. 19, 190-199. (doi:10.1016/j.cossms.2015.01.002) · doi:10.1016/j.cossms.2015.01.002
[6] Heimbs S. (2013) Foldcore sandwich structures and their impact behaviour: an overview. In Dynamic failure of composite and sandwich structures (eds Abrate S, Castanié B, Rajapakse YDS), Solid Mechanics and its Applications, vol. 192, pp. 491-544. Dordrecht, The Netherlands: Springer Science and Business Media.
[7] Schenk M, Guest SD. (2013) Geometry of Miura-folded metamaterials. Proc. Natl Acad. Sci. USA 110, 3276-3281. (doi:10.1073/pnas.1217998110) · doi:10.1073/pnas.1217998110
[8] Wei ZY, Guo ZV, Dudte L, Liang HY, Mahadevan L. (2013) Geometric mechanics of periodic pleated origami. Phys. Rev. Lett. 110, 215501. (doi:10.1103/PhysRevLett.110.215501) · doi:10.1103/PhysRevLett.110.215501
[9] Silverberg JL, Evans AA, McLeod L, Hayward RC, Hull T, Santangelo CD, Cohen I. (2014) Using origami design principles to fold reprogrammable mechanical metamaterials. Science 345, 647-650. (doi:10.1126/science.1252876) · doi:10.1126/science.1252876
[10] Waitukaitis S, Menaut R, Chen BGG, van Hecke M. (2015) Origami multistability: From single vertices to metasheets. Phys. Rev. Lett. 114, 055503. (doi:10.1103/PhysRevLett.114.055503) · doi:10.1103/PhysRevLett.114.055503
[11] Zhou X, Zhang S, You Z. (2016) Origami mechanical metamaterials based on the Miura-derivative fold patterns. Proc. R. Soc. A 472, 20160361. (doi:10.1098/rspa.2016.0361) · doi:10.1098/rspa.2016.0361
[12] Zakirov IM, Alekseev KA. (2010) Design of a wedge-shaped folded structure. J. Mach. Manufact. Reliab. 39, 412-417. (doi:10.3103/S105261881005002X) · doi:10.3103/S105261881005002X
[13] Wang K, Chen Y. (2011) Folding a patterned cylinder by rigid origami. In Origami5 (eds Wang-Iverson P, Lang RJ, Yim M), pp. 29-38. Boca Raton, FL: CRC Peters.
[14] Gattas JM, Wu W, You Z. (2013) Miura-base rigid origami: parameterizations of first-level derivative and piecewise geometries. J. Mech. Des. 135, 111011. (doi:10.1115/1.4025380) · doi:10.1115/1.4025380
[15] Lang RJ. (1996) A computational algorithm for origami design. In Proc. Twelfth Annual Symposium on Computational Geometry, Philadelphia, PA, 24-26 May, pp. 98-105. New York, NY: ACM.
[16] Tachi T. (2006) 3D origami design based on tucking molecule. In Origami4 (ed. Lang RJ), pp. 259-272. Natick, MA: A. K. Peters.
[17] Tachi T. (2013) Designing freeform origami tessellations by generalizing Resch’s pattern. J. Mech. Des. 135, 111006. (doi:10.1115/1.4025389) · doi:10.1115/1.4025389
[18] Zhou X, Wang H, You Z. (2015) Design of three-dimensional origami structures based on a vertex approach. Proc. R. Soc. A 471, 20150407. (doi:10.1098/rspa.2015.0407) · doi:10.1098/rspa.2015.0407
[19] Dudte LH, Vouga E, Tachi T, Mahadevan L. (2016) · doi:10.1038/NMAT4540
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.