JOURNAL OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES: J. IASS
POLAR METHOD TO DESIGN FOLDABLE PLATE STRUCTURES VALENTINA BEATINI1 1
Asst.Prof., Dept. of Architecture, Abant İzzet Baysal University, Turkey.
[email protected]
Editor’s Note: Manuscript submitted 5 December 2013; revisions received 11 August 2014 and 13 January 2015; accepted 21 April. This paper is open for written discussion, which should be submitted to the IASS Secretariat no later than December 2015.
ABSTRACT The paper considers non-developable foldable plate structures. It presents a design method to approximate any surface made by the translation of a generic curve along a straight line or connected segments. The method achieves a continuous, foldable 3D grid which approximates the target surface along two perpendicular directions. The parameters defining the target surface and the sharpness of the folded plates can be freely set. Keywords: folded plates, over constrained mechanisms, curved foldable plates, origami, kinetic architecture, flexible meshes, polar method 1. INTRODUCTION This paper is part of our research on foldable plate structures for architectural applications and presents a design technique, which we call the polar method. This kind of mechanisms has been used for solar panels in space engineering [1] and in some mechanical applications [2]. From the pioneering work on modular foldable paper by Resch [3], foldable plate structures are rapidly attracting interest for the design/packaging sectors [4] [5] [6]. Previous research highlighted very important although quite unsorted aspects of foldable plates. First of all, rigidity should be mentioned: this is the condition by which the movement occurs just on joints while plates act as rigid members and do not have to undergo any strain. Matrix models of the entire structure are discussed in [7] and [8]; while in [9] a computer program is presented to graphically compute rigid meshes in real time. Guarantee of an efficient use of the plates’ material is flat foldability, which means that in the folded configuration all faces lie in parallel planes. Local conditions (i.e. defined at each vertex) to ensure this property date back to the Maekawa and Kawasaki's Theorems [10]; indeed, they consider only arrangements where the sum of planar plates’ angles around a common vertex is 360°. Under such restriction, global flat foldability was discussed in [11].
Finally, developability had been addressed. It allows the mesh to be flattened onto a plane without distortion (compression or stretching); it is achieved when the Gaussian curvature of each inner vertex is zero [12] and it has especially been studied in origami area, since there the moduli come from a single continuous piece of paper. This quality is a sufficient, non-necessary, condition of an efficient use of the material. In [13] the good structural behavior of non-developable folded plates had been highlighted and the problem of achievable shapes has been initiated. Yet, a deep investigation of the design possibilities had been conducted so far only for developable folded plates [14]. The present paper contributes to this topic, and the polar method herein discussed allows approximating translational surfaces through a three-dimensional grid of non –developable folded plates. Specifically, when a continuous folded plate system is concerned, the method allows approximating any translational surface made by one generic planar curve moving along a straight line or a composition of segments. 2. PROPOSED DESIGN METHOD 2.1.
The system and its constraints
The method creates a three-dimensional foldable grid that can approximate a translation surface generated from two curves one of which is straight or composed of straight segments.
Copyright © 2015 by Valentina Beatini. Published by the International Association for Shell and Spatial Structures (IASS) with permission.
125
Vol. 56 (2015) No. 2 June n. 184
The simplest foldable system consists of four plates, Fig.1. All plates composing this system are quadrangular; they have two sides parallel to each other and no straight angles. Plates are organized in such structure that they join at each inner vertex of the grid, and hinged to each other along their sides. The planar angles of plates around each vertex are all equal to each other. From the above, the sum of planar angles of plates sharing an inner vertex is different from 360°. Therefore, plates cannot lie on one plane without detaching or deforming, i.e. the system is not developable. The arrangement of plates in fact generates a Kokotsakis mesh, which is a polyhedral structure consisting of a central polygon surrounded by a belt of polygons so that the relative motion between cyclically consecutive neighbor polygons is described by the motion of a spherical linkage. Kototwasky also demonstrated this through angular velocities analysis the foldability of a generic system of equal plates when the adjacent angles are either supplementary, or, as here, equal [15]. 2.2.
Design parameters
Let us consider a generic translational surface defined by means of two perpendicular curve named respectively Ti and Tj, where the curvature of Tj is zero at every point. The approximating system will reproduce the target surface through a 3d grid of plates. The target curve Ti therefore will be reproduced not by the single plates themselves, but by the projections of two adjacent strips of plates into their middle perpendicular plane parallel to the target curve. The same holds for approximating target curve Tj (Fig.2a). Putting together the achieved perpendicular projections, it is possible to design a double strip of plates along the generic curve Ti. Translating the plates along the perpendicular curve Tj provides the final grid. A minimum system of four plates sharing a vertex is fully defined by means of a set of six parameters. Under the design perspective, it is convenient to set the parameters that define the target approximation of the desired surface. Accordingly, we set for i=1,.., n-1 and j=1, …, m-1 (Fig.2b):
126
Figure 1: A system composed of four plates sharing a vertex and arranged into a non –developable unit
ai + ai +1 : span of plates Pi,j plus Pi+1,j along target curve Ti; a j + a j +1 : span of plates Pi,j plus Pi, j+1 along target
curve Tj; µi + µi +1 : central angle subtended by the chord covering interval i, i+1 on the target curve Ti. µ j + µ j +1 =0, central angle subtended by the chord covering interval j, j+1 on the curve Tj. Spans ai , ai , and angle µ i can vary along the relative curve, while angle µ j is null by definition of the target surface. In order to define the system completely, two more parameters need to be set, which relate with the three-dimensional arrangement of plates at the target configuration. These two parameters need to belong one per plane to their projections on planes parallel to the main directions of the target surface. Accordingly, they are chosen:
ϑ a j , j=1: half of the dihedral angle between plates Pi , j and Pi , j +1 on plane containing target curve Tj;
ϑai , i=1: projection of planar angle π ai , j = π ai +1, j which belongs to plates Pi,j and Pi+1,j on plane containing target curve Ti. Indeed, angle ϑai can assume any value in the range of 0°< ϑai
