Parametrisation and Application of Cube and Eggbox-type Folded

Parametrisation andApplication of Cube and

Eggbox-type FoldedGeometries

Ruikang Xiea, Yan Chena and Joseph M. Gattasb, *

aMotion Structure Laboratory, School of Mechanical Engineering, Tianjin University, Tianjin 300072, ChinabSchool of Civil Engineering, University of Queensland, St Lucia, QLD 4072, Australia

(Submitted on 20/10/2014, Reception of revised paper 30/06/2015, Accepted on 03/07/2015)

ABSTRACT: There exist several kirigami, or non-developable folded platepatterns that share the most useful properties of the widely-used Miura-oripattern, namely rigid-foldability, a tessellated unit geometry, and constituentelements composed of a single parallelogram plate. This paper first presentsparametrisations of five such patterns, two cube-type and three eggbox-type,consistent with that previously developed for Miura-type patterns. For eachpattern, the parametrisation establishes relationships between crease pattern,volumetric, and kinematic parameters. Along with the Miura-type patterns,these geometries form a set termed parallelogram-plate folded geometries andtogether they suggest a range of interesting structural engineering applications.This paper discusses two applications specifically enabled by the newparametrisations: space frame synthesis and hierarchical sandwich panels.

Key Words: rigid foldable origami, folded sandwich structures, kirigami

International Journal of Space Structures Vol. 30 No. 2 2015 99

1. INTRODUCTION1.1. Rigid tessellated origamiTessellation origami are a set of origami patterns thatconsist of a repetitive unit cell [1]. Rigid origami area set of origami patterns which consist of panels thatcan move continuously between folded states byrotating around crease lines without deformation [2].Origami patterns that are both rigid and tessellated aresimple to design and manufacture from rigidengineering sheet materials. They have thus been usedto develop many origami-inspired engineeringapplications, including deployable and spacestructures [3, 4, 5, 6, 7]. The Miura-ori pattern [8] andits derivative geometries [9, 10] are the most widelyused tessellated rigid patterns, with most researchinterest seen in their application as folded-core

sandwich panels [11, 12, 13, 14], cellularmetamaterials [15, 16, 17], and shelters [18, 19].Research in the application of other rigid tessellatedorigami families is developing rapidly, particularly inthe fields of smart materials and robotics [20].

1.2. Rigid Tessellated KirigamiKirigami patterns are similar to origami patternsexcept they can contain slits or punched out portions inthe sheet. They are thus non-developable, but can stillbe rigid-foldable and tessellated. Numerous rigidtessellated kirigami patterns have been identified assuitable for use as folded core patterns [21, 22, 23, 24]and early benchmarks of their performance suggeststhey can exceed the more widely-studied Miura-typefoldcore performance under out-of-plane impact loads

*Corresponding author e-mail: Joseph M. Gattas [email protected]

[25]. However existing pattern identifications vary andthere is no comprehensive parametrisations.Engineering adoption therefore is hindered.

This paper investigates five rigid tessellated kirigamipatterns that are composed of a repeated parallelogramplate: the planar cube, the inclined cube, the kirigamieggbox, the non-developable eggbox, and the arc-eggbox. Section 2 first presents a comprehensiveparameterisation for each pattern that relates creasepattern, kinematic, and volumetric parameters and isconsistent with Miura-ori parametrisations developedpreviously [9]. Section 3 then presents two applicationsenabled by the new folded plate geometries and relevantto structural engineering interests: space framesynthesis and hierarchical sandwich panels.

2. PATTERN PARAMETRISATIONSThe following section parametrises several of thecube-type patterns shown in [21]. For each, relationsbetween crease pattern, volumetric, and kinematicparameters are identified.

2.1. Planar CubeThe crease pattern for a planar cube pattern has threeparameters as shown in Figure 1(a): side length a,crease lines from the bottom m and crease lines fromthe left n. Valley folds are shown as dashed andmountain folds are shown as dot-dashed. The foldingmotion in the longitudinal x and lateral y direction isdefined with two dihedral angles A and B as shownin Figure 1(b). The vertex at the intersection of theith and jth crease lines is denoted by Vi, j, where i = 1,2, ... m, and j = 1, 2, ... n. A panel with vertices Vi, j,Vi + 1, j + 1 and Vi + 1, j is designated Pi, j.

In an unconstrained cube, these angles A and B aredecoupled and so the pattern possess multiple degrees-of-freedom. However if an assumption is made that alldiagonally-sequential panels, for example P1, 1 and P3, 3 or P3, 1 and P1, 3, remain parallel during folding,A and B are effectively coupled. The resultantgeometry is single-DOF and so can be controlled witha single parameter , see Figure 1(c).

The regular square crease pattern with uniform sidelength a and orthogonal i and j creases is in fact aspecial case of a parallegram geometry, shown inFigure 1(d). The general crease pattern is defined withsix parameters, m, n, and a as before, sector angle ,and side lengths b and h. Note that although it nolonger forms a cube when folded, for convenience thepattern is still referred to as the planar cube pattern.

It retains a single-DOF rigid-foldable mechanism togenerate a folded shape shown in Figure 1(e). A new

set of edge angle parameters, A, V, and M, now existwhich can be related to dihedral angle . A is relatedto as per eqn (2) in [9] and V and M relations arefound from the vector dot product of edges

and respectively:

(1)

(2)

(3)

The pattern therefore has a permissible folding rangefrom 0 V which corresponds to min where min is calculated with substitution into eqn (2):

(4)

Using the above parameters, the locations of allvertices in any folded configuration can bedetermined. In a 3D Cartesian coordinate system withorigin at V1, 1 and orientation shown in Figure 1(f), thecoordinate vector (x, y, z) of Vi, j can be given as:

(5)

(6)

(7)

where fI1, I2 is a helper logic function that relates twointegers I1 and I2 and their modulus after division is asfollows:

(8)

For example, j = 1, 2, 3, 4 gives fj, 2 = 0, 1, 1 , 2 and fj 1, 2 = 0, 0, 1, 1.

Figure 1(f) depicts the smallest tessellating unit ofthe planar cube pattern with m = n = 5. This unittessellates along axes xS and yS, where subscript Sdenotes skew axes, through vectors and

respectively. The inclination of axes xS

to x and yS relative to y is designated x and y,respectively. These can be related to existingparameters with triangle geometry as:

cos = sin cos cosA2 2

+ +cos = 1 (1 cos )[1 (1 cos )(1 cos )]V2

= + +cos 1 (1 cos )[1 (1 cos )(1 cos )]M2

cos = 11+cos

1min

=

+ +

y f b f h

f h

( cos )sin

sin(2 )

2(1 cos )

i i

j

, 2 1, 2

1, 2

= z f f f f h( ) ( ) sin sinj j i i1, 2 1, 4 1, 2 1, 4

f I I I0.5 mod ,I I, 1 1 21 2 ( )( )=

V V V V2, 2 2, 3 2, 2 3, 2

V V V V ,2, 3 2, 2 2, 3 3, 3

+ + x f a f h f b f h= cos ( )cosj, 2 j A i i1, 2 , 2 1, 2

V V1, 1 1, 5

V V ,1, 1 5, 1

Parametrisation and Application of Cube and Eggbox-type Folded Geometries

100 International Journal of Space Structures Vol. 30 No. 2 2015

Ruikang Xie, Yan Chen and Joseph M.Gattas


(9)

(10)

Volumetric length la, width lb, and height lh parametersare defined along axes xS, yS, and z, respectively. Thetotal volume V = lalblh sin(/2 x y) can be found

= +

+a/h

cot 2cos

sin(2 )(1 cos )cot( )x

=+

(b h)

b htan

cot

cosy

once length parameters are known. These can be relatedto existing parameters again with substitution into theabove vertex coordinates:

(11)

(12)

(13)

= l f a f h( cos )/cosa n, 2 n A x1, 2

= l f b f h( cos )sin /cosb m, 2 m y1, 2

=l hsin sinh

a)

1,1V 1,5V

5,1V 5,3V

3,5V

b)b)

x,n

y,m1,5V

1,1Vxyz

a

3,1V

1,3V

1,3V

g)

d) e)

x,n

y,m

h

h

a

b

f)la

lh

zxy

lb

M VA

c)

1,5V

1,1V

5,1V

5,3V 3,5V

xyz

1,3V3,1V

y

z5,1V

1,3V 5,3V

1,1V

x

z1,5V

3,1V 3,5V

1,1V

x

z1,5V

3,1V 3,5V

1,1VA

y

z 5,1V1,3V

5,3V

1,1VB

3,5V5,1V

5,3V

3,1V

V

xxyz y

xy

S S

AB

2,3P

Figure 1. Planar cube pattern a) crease pattern; b) isometric and elevations of multi-DOF folding motion; c) isometric andelevations of 1-DOF folding motion. General pattern form d) crease pattern; e) kinematic parameters; g) volumetric parameters;

g) comparison of simulated and prototype folding motion.

To summarise, fifteen parameters have been definedon the planar cube pattern: constant parameters a, b, h,, m, and n; and variable parameters , A, V, M, x,y, la, lb, and lh. Eight relations exist amongst them,eqns (13) and eqns (913). Seven independentparameters are therefore sufficient to uniquely define aplanar cube pattern. Good correlation is seen betweensimulated and prototype folding motion for a cardprototype with a = 10 mm, b = 20 mm, h = 30 mm, = /2, m = n = 6, and /2, Figure 1(g).

2.2. Inclined CubeIn a planar cube unit geometry, panels P1, 1 and P3, 3rest on the z = 0 plane and panels P1, 3 and P3, 1 on thez = lh plane. This occurs from the mountain-valley orM-V assignation at the edge creases. Note that valleyfold nomenclature V is distinct from vertexnomenclature Vi, j. The four edge creases of the first

and second pair of panels have assignations V-V-V-Vand M-M-M-M, respectively, which is to say all valleyor all mountain edge creases. A cube pattern variantexists where these M-V assignations are altered to V-M-M-V and M-V-V-M, as shown in Figure 2(a). Thiscauses P3, 3 to lie on the z = 2lh plane. Such a pattern istermed the inclined cube pattern.

If an assumption is again made that diagonallysequential panels remain parallel during folding,single-DOF rigid-foldability is achieved and theinclined cube crease pattern can be defined with sevenparameters: six crease pattern parameters m, n, a, b, h, and a variable parameter. Relations between , A,and V are still given by eqns (12), shown in Figure2(b). M is given by:

(14) = + + cos (1 cos )[(1 cos )(1 cos ) 1] 1M2



3,1V

1,3V

a)

1,1V

b)

x,n

y,m

h

h

1,1Va

b

d)

3,1V

1,3V3,1V

f)

1,1V

1,1V y xz

1,1V

e)

MV

A

1,3V

1,3V

xyy x

x

yyx

zz

1,3V3,1V

x

zz xzz x

c)

xyzxyS S

y xz

dm

la

lh

lb

Figure 2. Inclined cube pattern parametrisation a) unfolded configuration and crease pattern parameters; b) folded configurationand kinematic parameters; c) original unit volume; d) inclined orientation of orthogonal pattern; e) inclined orientation of skew

pattern; f) comparison of simulated and prototype folding motion.



The similarity between base and inclined cube patternsmeans vertex Vi, j coordinate x and y components arestill given by eqns (56). The z component is given by:

(15)

The retention of the (x, y, z) coordinate system fromthe planar cube is convenient for rapid vertexparametrisation, however not for representation of theminimum bounding volume. The inclined cube patternhas a bounded ceiling plane through vertices V1,1, V1, 3,and V3, 1; and along axes xS and yS as shown in Figure2(c). Similarly, a floor bound exists though vertices V2, 2, V1, 4, and V4, 1. To enable calculation of distancebetween these two planes and thus volumetricparameters, the z axis is defined as orthogonal to the xS yS plane, where the superscript denotes atransformed Cartesian coordinate system, see Figure2(d). Axis y is defined as perpendicular to z andparallel to line Axis x is orthogonal to z and

y and through V1, 1. Axes x y are therefore inclinedat angle to axes x y, with given by:

(16)

where k = b + h(1 cos (1 + cos )). Axis x isperpendicular to axis y and so is perpendicular to line

Length dm is defined in the x y plane as the

perpendicular distance from V1, 1 to line :

(17)

where x is given by eqn (9). Axis z is inclined atangle relative to the x y ground plane and can becalculated as:

(18)

Thus, vertex coordinates Vi, j in the (x, y, z) coordinatesystem can be transformed to coordinates Vi, j in the(x, y, z) coordinate system with sequential 3Drotation matrices Ry and Rz as follows:

(19)

with coordinate components given directly as:

(20)

=

h

dtan

sin sin

m

V R R V( ) ( )i j y z i, j, =

=

+

a k h

(k htan

cos cos

(1 cos ))sin

da h

=( cos )

coscos( )m

A

xx

V V .1, 3 3, 1

V V .1, 3 3, 1

= + z (f f h) sin sinj i1, 2 1, 2

V V .1, 3 3, 1

= + +x x y zcos ( cos sin ) sin

(21)

(22)

This transformed system allows straightforwardcalculation of inclined volumetric parameters l a, l b,and l h. These are shown in Figure 2(e) and can becalculated from:

(23)

(24)

(25)

where la and lb are given by eqn (11) and eqn (12),respectively, xs is the inclination angle between axesxS and xS, and ys is the inclination angle between axesyS and yS. These are given by:

(26)

(27)

where y is given by eqn (10). Good correlation isseen between simulated and prototype folding motionfor a card prototype with a = 10 mm, b = 20 mm, h = 30 mm, = /2, m = n = 6, and /2,shown in Figure 2(f).

2.3. Kirigami EggboxThe kirigami eggbox is a rigid foldable pattern shownin [21] that is notable for engineering use because ofits similarity in form to expanded metal laths.Additionally, in a fully-folded configuration, thepattern forms an eggbox-like tessellation with apexvertices bounded by three panels. It has diamond-shaped voids in a crease pattern as shown in Figure 3(a) and can be considered as a variant of theinclined cube pattern. In the developable eggbox, V1, 2 V2, 2 V2, 3 = and V1, 3V1, 2V2, 2 = whichis vice versa to the inclined cube.

The pattern can therefore be parametrised with m, n,a, b, h, , and a variable parameter such as . Relationswith A, V, and M, shown in Figure 3(b), can bederived in a manner similar to that used for cube-typepatterns, to give:

(28)

(29)

(30)

= +cos sin cos cosA2 2

= + cos 1 (1 cos )[1 (1 cos )(1 cos )]v2

= tan cos( ) tanxs x

= +tan sin( ) tanys y

=l l /cos yb b s

l a b b[( cos )cos sin sin ]sinh = + +

=l l /cos xa a s

= + +z x y zsin ( cos sin ) cos

y x ysin cos = +

=M V

which gives a fully-folded lower bound at :

(31)

The similarity between developable eggbox andinclined cube patterns means vertex Vi, j coordinate zcomponent is still given by eqn (15). The x and ycomponents are given by:

(32)

(33)

These can be transformed to inclined coordinatesVi, j in the (x, y, z) with sequential rotations as givenby eqns (1922). However the rotational parameters toachieve this are altered to:

(34)

(35)

(36)

(37)

where k = b h (1 + cos (cos 1)).Inclination parameter is then still given by eqn (18).

Similarly, eqs. (24), (25) and (27) are still valid for l b, l h,

= + x f a f h f b f hcos ( )cosj j A i i, 2 1, 2 , 2 1, 2

y f b f h

f h

( cos )sin

sin(2 )

2(cos 1)

i i

j

, 2 1, 2

1, 2

=

+

cos 11

1 cosmin

=

+

a k h

k htan

cos cos

( (1 cos ))sin

=

+

=

a hcot 2 / cossin(2 )(1 cos )

cot ( )x

=

b h

b htan

( )cos

cosy

=

+da h cos

coscos( )m

A

xx

and ys respectively. However the following relations areused for la and xs:

(38)

(39)

Good correlation is seen between simulated andprototype folding motion for a card prototype with a =b = 20 mm, h = 30 mm, = 65, m = 6, n = 6, and 1.27, see Figure 3(d).

2.4. Non-Developable EggboxWhen fully-folded, the cube-eggbox has apex verticesbounded by three panels. Another rigid-foldablepattern exists with a similar eggbox-like form but withapexes bounded by four panels. It is a well-studiedpattern [22, 23] and is known to be non-developableand composed of the crease pattern shown in in Figure4(a). The folded configuration is formed by connecting

adjacent zigzag creases, for example edges

and It can be seen that the crease pattern

can be completely determined by five constantparameters: side lengths a and b, sector angle , andstraight and zigzag crease lines m and n. Angularparameters including dihedral angles A and Z andedge angles A and Z are shown in Figure 4(b). Usingmethods described for the Miura pattern in [9], thefollowing relations can be found to exist betweendihedral angles:

(40)

(41)

=

lf a f h cos

cos cosan n A

x xs

, 2 1, 2

= +tan cos( ) tanxs x

V V1, 2 2, 2

V V .1, 3 2, 3

+ + =(1 cos )(1 cos ) 4 cosZ A2

= +cos sin cos cosZ A2 2



b)

c)

a)

y xz

x,n

y,m

h

h

a

b

1,1V

3,1V

1,3V

d)

M

A

V

lb

lh

la

Figure 3. Kirigami eggbox pattern a) crease pattern; b) folded configuration and kinematic parameters; c) volumetric parameters;d) comparison of simulated and prototype folding motion.



The relation for A described in eqn (28) can also beshown to remain valid with Z substituted for .

The locations of all vertices Vi, j with origin at V1, 1and with orientation shown in Figure 4(c) are given bythe coordinate vector (x, y, z) as:

(42)

(43)

(44)

where gI1, I2 is equal to I1 modulo I2. Volumetricparameters can be determined from nodal coordinatesas:

(45)

(46)

(47)

Six parameters are therefore independent and canuniquely define a non-developable eggbox geometry.Folding motion for a pattern with a = b = 20 mm, = 70, m = 5, n = 7, and A 0 is shown inFigure 4(d).

2.5. Arc-EggboxA non-developable eggbox pattern with differentsector angles on alternate zigzag creases forms acurved arc profile, similar to the Arc-Miura pattern in[9]. The unfolded unit geometry is shown in Figure

= x j a( 1) sin( /2)A

= y i b( 1) sin( /2)Z

= + z g a g bcos( /2) cos( /2)j A i Z1, 2 1, 2

= l n a( 1) sin( /2)a A

= l m b( 1) sin( /2)b Z

= +l a bcos( /2) cos( /2)h A Z

5(a). Sector angles 1 and 2 correspond to mountainand valley ridges on alternate zigzag creases,respectively, where 1 < 2. Pattern panels thereforehave four different side lengths, a1, a2, b1 and b2, witha1 > a2 and b1 > b2, and two sector angles, 1 and 2.Among these six parameters, four are independent, asthe following two relations can be found:

(48)

(49)

Alternate crease pattern edges can be joined tomake a folded configuration of the arc-eggbox pattern,shown in Figure 5(b). The A, Z, and Z angularparameters of the non-developable eggbox pattern aredifferent for mountain and valley apexes in the arc-eggbox pattern. The parameter A however is commonacross the whole pattern, so in total there exist sevenangular parameters. At mountain apexes, eqn (28) andeqns (4041) can be formulated with A and mountainparameters MZ, MA, MZ and 1, where the subscriptM denotes a mountain parameter. Similarly, eqn (28)and eqns (4041) can be reformulated with A andvalley parameters VZ, VA, VZ and 2. This gives sixrelations amongst seven parameters and so a singlevariable is sufficient to find remaining values.

The projection of arc-eggbox vertices in the x zplane forms a series of arc profiles, shown in Figure 5(d).Along odd j creases, nodes lie along inner and outerarcs, RV, i and RV, o, respectively. Similarly, along evenj creases, nodes lie along inner and outer arcs, RM, i

b bsin sin1 1 2 2 =

a b a bcos cos2 1 1 1 2 2 + = +

b)a)

a

b

3,1V1,3V

c)

x,n

y,m

1,1V 1,2V

3,1V

AZxy

z1,3V

1,1V

1,2V3,1V

ZA

la

lh

lb

d)

Figure 4. Non-developable eggbox pattern a) crease pattern; b) folded configuration and kinematic parameters; c) volumetricparameters; d) comparison of simulated and prototype folding motion.

and RM, o, respectively. Sequential j creases areseparated by polar angle . All parameters can berelated to existing parameters from triangle geometryto give:

(50)

(51)

(52)

(53)

(54)

Any vertex in a folded arc-eggbox pattern vi, j can bederived in a 3D cylindrical coordinate system where(x, y, z) = (r cos , y, r sin ) with orientation shown inFigure 5(c). The three components (r, , y) of Vi, j canbe given as:

(55)

(56)

(57)

where c1 = RV, o/RV, i 1 and c2 = RM, i/RV, i 1. Simulatedfolding motion for a pattern with a2 = b2 = 20 mm,

= =R a a a Rsin( /2)/sin /V o MA V i, 1 1 2 ,

=R Rsin( /2) / sin( /2)M i VA MA V i, ,

=R Rsin( /2)/ sin( /2)M o VA MA V o, ,

= ( )/2VA MA

=R a sin( /2)/sinV i MA, 2

r R c g c g c c g g(1 )V i j i i j, 1 1, 2 2 1, 2 1 2 1, 2 1, 2= + + +

j( 1) =

= y i b( 1) sin( /2)VA2

1 = 55, 2 = 70, m = 5, n = 7, and 0 A is shownin Figure 5(e).

2.6. DiscussionThe common parametrisation between above cube andeggbox geometries, paired with previous Miura-baseparametrisations [9], allows for easier application offolded plate geometries to engineering design. Mostdirectly, folded cores with the same effective densitiescan de designed and compared when used asspacefilling folded core structures as originallysuggested for several of the above geometries by [21].Instead of designing for a common density, differentstructures can be designed that possess the samevolume of material, as the first four of the abovepatterns and the Miura pattern form a set of geometriesthat can be constructed from identical parallelogramplates. For example, Figure 6 shows the fivegeometries that are each constructed from 16 plateswith a = 300 mm, b = 300 mm, and = 70. Similarequivalence exists between Arc-Eggbox and Arc-Miura patterns. Single-curved cube patterns are knownto exist [21], however they possess multiple degrees-of-freedom and cannot be constructed from a repeatedplate size, so are not considered in the present work.

3. APPLICATION3.1. Space Frame SynthesisBeyond the ability to translate between differentfolded plate architectures, several applications are now



b)a)

a2

3,1V1,3V

c)

x,n

y,m

2

1,1V 1,2V

3,1V

MAVZ

1,3V1,1V

3,1V

e) yR

RV,o

MZAb2a1

b11 RV,i

RM,iRM,o

MZ

VA

VZ

d)

RM,o

RV,i

Figure 5. Arc-eggbox pattern a) crease pattern; b) folded configuration and kinematic parameters; c) volumetric parameters; d)projection showing vertex radii; e) comparison of simulated and prototype folding motion.



possible with the use of cube and eggbox-typeparametrisations. For reasons discussed below, theseare not possible with the more widely-known Miura-type pattern. One such application that is perhaps ofparticular interest to many researchers is the synthesisof space frame structures from folded plategeometries. The cube and eggbox-type geometrieseach have convex vertices and so a variety of spaceframe architectures can be generated directly byconsidering crease lines as bars.

For example, Figure 7(a) shows two-way by two-way, tetrahedral, and octahedral space frames. The firstis generated from the creases of a planar cube patternwith a = b = 0, h = 10 mm, = /3, m = n = 10, and = 135. The second is generated from the creases ofan inclined cube pattern with parameters a = b = h = 10 mm, = /3, m = n = 5, and = 91. The third is

generated from an ND-Eggbox pattern withparameters a = b = 10 mm, = /3, m = n = 5, and = 109.5. Note that the creases of the inclined cubeand non-developable eggbox pattern form triple-layerframes, however only the lower half of creases, that isto say corresponding to a two-layer tetrahedral oroctahedral frame, are shown for clarity.

Manufacturing methods and multifunctionalexamples of these space frame typologies are alreadyutilised in applications across many fields [26, 27].The use of folded plate vertices however allows high-degree parametric control over space frame geometrieswith crease pattern side length parameters equivalentto space frame element lengths. Additionally, thesynthesis can be applied to piecewise folded plategeometries to form more complex global curvatures.Piecewise folded geometries are simple to design by

Figure 6. Equivalent-area geometries from left to right: planar cube, inclined cube, kirigami eggbox, non-developable eggbox,and Miura-ori.

a) b)

Figure 7. Space frame synthesis with folded plate edges. From left to right: a) planar cube, inclined cube, ND-eggbox, b) piecewise ND-Arc-Eggbox (plates), piecewise ND-Arc-Eggbox (frame). From top to bottom: pattern crease elements only,

addition of top/bottom elements, plan, front elevation.

enforcing common edge lengths and angles [9]between non-similar patterns. An example piecewisegeometry with a non-developable eggbox and two arc-eggbox patterns is shown in Figure 7(b). Thesynthesised space frame is also shown, again with thetop-half removed for clarity. The stability of thesestructures is yet to be considered carefully and is likelyhighly application dependent. If face-plane elementsor nodal connections are not strong enough to lock theDOF, other methods should be introduced such asadding bars or increasing the thickness of stack layer.

3.2. Hierarchical Sandwich PanelsHierarchical structures can exhibit superiormechanical properties to continuum structures as theycan be modified at more than one length scale [28, 29].Hierarchical sandwich structures have been proposedpreviously for single-corrugated and honeycomb coretypes [30, 31, 32] and numerous new forms can begenerated with folded patterns explored in this paper.

Cube-type geometries possess a parallelogrammicfolded volume. In origami-based engineeringapplications, such a shape is common as it is the

constituent component of thick-panel Miura-ori. A cube-type geometry can therefore be designed to filla target thick-panel Miura plate and so enable theformation of hierarchical folded core sandwich. Anexample in Figure 8(a) shows the equivalent-volumeplanar cube pattern corresponding to a single thick-panel Miura plate with parameters a = b = 200 mm, = 66, and a plate thickness of 21 mm. Also shown isa hierarchical arrangement in which a global Miura-typefolded core is assembled from local cube-type foldedcore sandwich panels. Similarly, planar cube and non-developable eggbox geometries can form a rectangularcuboid volume, which is the constituent component ofthick-panel cube-type patterns with a sector angleparameter = /2. Two hierarchical sandwich panelsare shown in Figure 8(b)(c) in which the same globalcube pattern is formed from local non-developableeggbox and planar cube panels, respectively.

4. CONCLUSIONThis paper has presented parametrisations for twocube-type and three eggbox-type non-developablefolded plate geometries. For each pattern, relationships



Figure 8. Hierarchical folded sandwich structures. a) Miura/Cube (global/local); b) cube/non-developable eggbox; c) cube/cubesandwich architectures.

a)

b)

c)



between crease pattern, volumetric, and kinematicparameters are presented. Pattern vertex coordinatesare also given, allowing straightforward simulatedfolding motion which has been validated bycomparison with physical prototype folding motion.The development of a consistent parametrisationallows for the easier adoption to engineeringapplications and translation with existing Miura-typepatterns. All parametrisations presented in this paperare also compiled into a MATLAB Toolbox freelyavailable for research purposes, which can bedownloaded from [33].

Additionally, two applications enabled by cubeand eggbox folded plate geometries were presented.The first demonstrated the synthesis of space framearchitectures directly from folded pattern creases.The second demonstrated the ability to producehierarchical folded sandwich structures with theextended range of folded geometric envelopes. Theusefulness or feasibility of these applications is yet tobe determined, but they demonstrate potential waysin which folded plate geometries can be used forparametric structural design.

5. ACKNOWLEDGEMENTSY. Chen would like to thank the financial support fromthe National Natural Science Foundation of China(Projects No. 51275334, No. 51290293, and No.51422506). J.M. Gattas would like to thank theConfucius Institute and Faculty of Engineering,Architecture, and IT at the University of Queenslandfor the Hanban Visiting Fellowship.

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Parametrisation and Application of Cube and Eggbox-type Folded