the snap-back behavior of a self-deploying cylinder based on origami with kresling pattern

Proceedings of the IASS-SLTE 2014 Symposium “Shells, Membranes and Spatial Structures: Footprints”

15 to 19 September 2014, Brasilia, Brazil Reyolando M.L.R.F. BRASIL and Ruy M.O. PAULETTI (eds.)

Copyright © 2014 by the authors.

Published by the International Association for Shell and Spatial Structures (IASS) with permission. 1

The Snap-back Behavior of a Self-deploying Cylinder Based on

Origami with Kresling Pattern

Jianguo CAI*, Jin ZHANGa, Ya ZHOU

a, Jian FENG

a

*School of Civil Engineering, Southeast University

Nanjing, 210096, China

Email: [email protected]; [email protected]

aSchool of Civil Engineering, Southeast University

Abstract

The deployment of a cylinder based on origami with Kresling pattern, whose basic mechanisms are formed by

the buckling of a thin cylindrical shell under torsional loading, is studied in this paper. The model consists of

identical triangular panels with cyclic symmetry and has a small displacement internal inextensional mechanism.

Firstly, geometric formulation of the design problem is presented. Then assuming that the deployment and

folding process is uniform, the bistable behavior of the cylinder is discussed. It can be found that, during the

deployment, the dimensionless strain energy increases firstly and then reduces to zero but followed by another

sharp increase. Moreover, the limit condition of geometry parameters for the bistable phenomenon is also

discussed. Finally, the bistable behavior is also studied by using numerical simulations for simple and more

complex case of the cylinder with multistory. The numerical results agree well with the analytical predictions.

Therefore, comparisons with finite element predictions have shown that the analytical solutions given in this

paper are accurate and have validated the assumptions made in the derivations.

Keywords: Origami, deployable structure, bistable behavior, snap-back behavior

1. Introduction

Origami, the Japanese and Chinese traditional paper craft, has been proved as a valuable tool to develop various

engineering applications in numerous fields [1]. Dr. Nojima Taketoshi have developed several patterns, which

ranges from environmentally friendly containers to medical applications, from vehicle parts to new insulation

material configurations, from robotics to education [2].

Deployable structure, a structure that can change its size by changing its shape, is widely used in daily life such

as tents and umbrellas. Current interest in deployable structures arises not only from their potential in space but

also many other areas [3-6]. Several designs of origami structures have been proposed for deployable structures

from around 1970s, such as the Miura-ori, which is a well-known rigid origami structure utilized in the

packaging of deployable solar panels for use in space or in the folding of maps [7]. Miura-ori provides a one-

degree of freedom (DOF) mechanism from a developed state to a flat-folded state.

One of the important problems on deployable structure is to fold a cylindrical tube in axial direction to a flat

state, while keeping its axis and internal envelope like a bellow. The best way to understand the background of

this problem is to consider the case of post buckling behavior of a thin cylindrical shell under axial loading. The

most famous folding pattern is the Yoshimura or diamond pattern [8-10]. But during the folding in the axial

direction, significant in-plane stretching occurs for the Yoshimura pattern [11]. Then Guest and Pellegrino

proposed a variation of cylindrical foldable shell by twisting Yoshimura-pattern [12-14]. This model consists of

identical triangular panels on a helical strip and has small-displacement internal inextensional mechanisms.

Tachi and Miura suggested some cylindrical deployable structures in which every element of the surface is

geometrically free of distortion [15-17]. This enables mechanisms with stiff materials that can potentially used to

design repeatedly foldable structures.


Copyright © 2014 by the author(s).

Published by the International Association for Shell and Spatial Structures (IASS) with permission.

On the other hand, if the thin cylindrical shell is under torsional load, another folding pattern, named “Kresling

pattern”, was obtained by Kresling[18-20]. He gave a simple experiment in Ref. [20] that a thin-walled sheet is

wrapped around two coaxial mandrels, leaving a gap. When the mandrels are twisted, highly regular folding

pattern appears as shown in Fig.1. Hunt and Ario [21] have studied the twist buckling pattern to consider its

accommodation to fold to a flat diaphragm. The critical and initial post-buckling effects were also investigated

through concepts of energy minimization. It should be noted that the Kresling pattern and the pattern studied by

Guest and Pellegrino [12-14] are similar. For the latter one, triangulated cylinders made from identical triangular

plates arranged on helical strips, a general description can be based on the observation that the nodes of the

triangulated cylinders lie on the intersection of three sets of helices. When one helix becomes several parallel

circles, the Kresling pattern is obtained. In this paper, the geometric design of a cylindrical shell based on the

Kresling pattern is firstly studied. Then the mechanical behavior, especially the bistable behavior, will be

investigated analytically. Furthermore, a numerical study based on ABAQUS will be used to validate the

analytical results.

Figure 1: Kresling pattern of thin cylindrical shells [20]

2. Geometry design of the kresling pattern

When designing the Kresling pattern of a thin cylindrical shell, we start with a flat strip paper as shown in Fig.

2(a). As described by Guest and Pellegrino [12-14], it may have small-displacement internal inextensional

mechanism. Then the fully folded configuration, shown in Fig. 2(b) is selected as geometrical compatibility.

After that, the structure can be deployed to the configuration shown in Fig. 2(c).

Figure 2: The construction process and the deployable process

If the length of the strip l, the height h and the number of the elements n are given, then we need to find the

angles between the diagonal fold lines and the horizontal line β and γ. All the geometric parameters are given in

Fig.3. It should be noted that in this paper every element is identical. That is to say the structure is cyclic

symmetry. Then it can be found the polygon formed by the upper lines or lower lines of the paper strip in the

fully folded configuration is regular polygon.

Figure 4 shows the movement of the fold line between the adjacent elements. The angle ∠1′06 can be obtained

as




1'06 π 2φ

(1)

The lines 1′3 and 35′, as shown in Fig.5, are two side of the regular polygon formed by the upper lines of the

paper strip. The angle relations can be obtained from Fig.5 as

1'06 1'30 31'0 1'30 460 1'30 635 1'30 5'36 1'35'

(2)

Figure 3: Geometric parameters of the flat paper strip

(a) deployed state (b) folded state

Figure 4: The movement between the adjacent elements

Figure 5: The sides of the regular polygon formed by the upper lines of the paper strip

Then the angle ∠1′35′ is

1'35' π 2φ

(3)

The sum of the inner angles of the regular polygon formed by the upper lines of the paper strip can be given as

1'35' π 2φ 2 πn n n

(4)

which leads to

πφ

n

(5)

The length of upper or lower lines of every element is a, which is equal to l/n. As shown in Fig.3, draw a line 2G

from point 2, which is perpendicular to line 14. Assuming the length of line 2G is d, in triangles 12G and 24G, it

can be obtained that

1 cotφGl d (6)




2 2 2 2

4 ( / )Gl a d l n d (7)

where l1G and l4G are the lengths of line 1G and 4G.

The area of the triangle 124 can be given as

2 14

1 1

2 2Ga h l l (8)

Substituting Eqs. (6) and (7) to Eq. (8) leads to

2 2( cotφ )ah d d a d (9)

The length d can be obtained by Eq. (9). Then the angles β and γ can be given as

γ arcsin arcsind nd

a l (10)

1β π-φ-γ= π arcsin

n nd

n l

(11)

3. Snap-back behavior

The movement process and the bistable behavior of the thin cylindrical shell based on Kresling origami are

studied in this section. As shown in Fig.6, during the motion the upper regular polygons formed by the upper

lines AB, BC, CD… have a relative rotation, θ, to the lower regular polygons formed by the lower lines 12, 23,

34… The radius of the circumscribed circle of these two regular polygons, R, is

2

sin

a

R

n

π

(12)

Figure 6: The relation between two regular polygons formed by the upper and lower lines of the paper strip

Figure 7: The cylindrical shell during the motion

http://dict.cn/circumscribed%20circle




We denote the angle between the line A2 and the horizontal plane is δ, which is shown in Fig.7, and the height of

the cylindrical shell during the motion is

sinH b (13)

If the center of the lower regular polygon is the origin of the cylinder coordinate system, points A and 2 have

coordinates (R, θ, b sinδ) and (R, 2π/n, 0), respectively. Then the distance between A and 2 is given as

2 22 /[2 sin( )] ( sin )

2

nb R b

(14)

The relation between angles θ and δ is obtained as

2 cos2arcsin

2

b

n R

(15)

It can be seen from Eq. (15) that

cos1

2

b

R

(16)

Substituting Eq. (12) into Eq. (16) leads to

1

cos sin

b

a

n

(17)

If the cylindrical shell must be moved to the fully folded configuration, Eq. (17) becomes

1

sin

b

a

n

(18)

The coordinate of point 3 is (R, 4π/n, 0), and hence the distance between A and 3 is given as

2 2cos[2 sin( arcsin )] ( sin )

2

bc R b

n R

(19)

which leads to

2

21

sin arcsin cos sin sin

sin

c b b

a n a n a

n

(20)

In the fully folded configuration (δ=0), Eq. (20) becomes

1sin arcsin sin

sin

c b

a n a n

n

(21)

The relations between the length ratio c/a and b/a of different element numbers are given in Fig. 8. It can be

found that the length ratio c/a increases firstly and then reduces with the increase of the length ratio b/a. It is also

be noted that the highest values of b/a and c/a are identical, which are sqrt(2), 2 and 1/sin(π/10) for element

number 4, 6, and 10, respectively. Furthermore, the highest value of b/a and c/a also increases with the increase

of the element number.

Then we will study the elastic strain of the truss model of the cylindrical shell during the motion. In this study,

the lengths a and b are assumed to be constant, i.e. assuming that all strain can be modeled in terms of variable

c only. With these hypotheses, the folding properties of the thin triangulated cylinder can be investigated most

effectively on a plot of c/a versus δ. A special case with n=6 is considered here. Then Eq. (20) can be written as

2 2

2sin arcsin cos sin6 2

c b b

a a a

(22)




Figure 8: Plots of c/a against b/a

(a) (b)

Figure 9: Plots of c/a versus δ

For simplicity, only the hypothesis of linear elastic strain will be investigated in this analytical approach, which

is defined as

0 0

0 0

/ /

/

c c c a c a

c c a

(23)

where c0 is the length of c at the folded configuration (δ=0). Then the dimensionless strain energy of unit length

can be defined as:

21

2

Ww

E (24)

where E is Young’s Modulus of the elastic bar.

Figure 9 shows a plot of c/a versus δ for triangulated cylinders with different values of b/a. The dimensionless

strain energy of unit length w of bar during the motion is given in Fig.10. It can be seen from Fig. 9(a) that for

the case b/a=0.5, the length c decreases with the increase of δ. That is to say the strain ε is increasing during the

deployment of the cylinder. It should be noted that any given value c/a in Fig.9 is corresponding to a unique

configuration of the cylinder. The other curve in Fig. 9(a), for the case b/a=1.0, shows similar behavior.

For the curves shown in Fig. 9(b), c/a initially increases with δ and then, having reached a maximum, starts

decreasing. During the deployment (with the increase of δ), the dimensionless strain energy of unit length w

increases firstly and then reduces to zero. After that, it also increases significantly. For the case b/a=1.5, the

cylindrical shell with 2.0616>c/a≥1.9605 have two different and totally unstrained configurations. That shows

the cylinder is a bistable structure. In particular, c/a=1.9605 gives one configuration in the fully folded state

(δ=0) and the other one when δ=1.3039. Figure 9 also gives some special cylinders whose c/a is an analytical

maximal value. For the case b/a=1.5, it occurs when δ=0.8411 and hence c/a=2.0616. The information for the

cases b/a=1.2, 1.1, 1.01 is also given in Table 1. It can be found that the gap between the upper and lower limit

of c/a, which is corresponding to two totally unstrained configurations, reduces with the decreases of b/a. For the

case b/a=1.01, the gap is 0.00003. Then when decreases to 1.0, the bistable phenomenon disappears. Therefore,

http://dict.cnki.net/dict_result.aspx?searchword=%e7%8e%b0%e8%b1%a1&tjType=sentence&style=&t=phenomenon




it can be concluded that when the element number n=6, the value b/a of for the cylindrical shell with bistable

behavior should satisfy

1 2b

a (25)

(a) (b)

Figure 10: The dimensionless strain energy of unit length during the motion

Table 1: The typical value of c/a and δ during the motion

b/a

δ c/a

With the largest

value of c

With the minimal

energy

With the largest value

of c

With the minimal

energy

1.01 0.1408 0.1996 1.73784 1.73781

1.1 0.4296 0.6181 1.7916 1.7878

1.2 0.5857 0.857 1.8547 1.8392

1.5 0.8411 1.3039 2.0616 1.9605

4. Numerical verification

With the hypothesis that a and b remain constant during the motion, while c varies, preliminary estimates of

strains during the motion are investigated in the previous section. More detailed estimates of strains are available

in this section. The deployment and folding of the cylinder were simulated with nonlinear finite element analysis

calculated with ABAQUS. To compare with the analytical results, the truss elements were used for simplicity.

Moreover, to model the constant value of a and b in the simulation, their Young’s Modulus was defined at least

two orders higher than the one of elastic bars c. To monitor the complete movement path of the cylinder,

displacement control is necessary because of the existence of instability phenomenon. The analysis was carried

out with displacement control, by defining vertical boundary at the nodes of the top polygonal panel and

displacing it with the height of the cylinder with small increments. The nodes of the bottom polygons were

constrained both in the radial direction and in the vertical direction. In this numerical example, b=1500 mm,

a=1000 mm, the area of bar is 393 mm2.

(a)the folded configuration (b) the deployable configuration

Figure 11: The deployment of one segment




4.1. Deployment process

The initial configuration of the structure is a fully folded shape with δ=0 when c=1960.4 mm as shown in Fig.11

(a). The displacement applied in the nodes of the top polygonal panel is 1500 mm. The final deployable

configuration is given in Fig.11(b). Furthermore, the energy during the motion is also shown in Fig.12.

Figure 12: The energy during the deployment of one segment Figure 13: The folding process

It can be seen from Fig. 12 that the shape of the curve is similar as the analytical result given in Fig. 10(b). The

cylinder also has the two unstrained configurations. Except the fully folded configuration, the other one is in the

state when δ=1.3035, while the analytical result is 1.3039. The highest energy during the bistable phenomenon

occurs when δ=0.8413, while the analytical result is 0.8411 shown in Table 1. It can be found the analytical

results have good accuracy.

4.2. Folding process

In this section, the folding process of the cylinder is investigated. The state with δ=1.286 is chosen as the initial

configuration. At this time, the height of the cylinder is 1440 mm. Then the displacement applied in the nodes of

the top polygonal panel is -1440 mm. The folding process is shown in Fig.13. And the initial and final

configurations during the motion are shown in Fig.14.

(a)the initial configuration (b) the flat configuration

Figure 14: The folding process of one segment

The energy during the folding process is given in Fig.15. It can be seen that the cylinder also has the bistable

behavior. This is because the value of c/a is larger than 1.9605. Except the initial configuration with δ=1.286, the

other one is near the fully folded configuration δ=0.1484. The highest energy during the folding process occurs

when δ=0.8412, while the analytical result is 0.8411 shown in Table 1.

The motion of the cylinder with multi segments is also studied. The energy during the motion is similar as that

for the one segment as shown in Fig. 15. The movement of the multistory structure is given in Fig.16.

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Figure 15: The energy during the folding process of one segment

(a) (b) (c)

(d) (e) (f)

Figure 16: The folding process of the cylinder with three segments

5. Conclusions

This paper addresses an interesting bistable phenomenon in the axial tension or compression of the thin

cylindrical shell with Kresling pattern. Firstly, the geometry design of this cylinder was investigated. When the

geometry of a paper strip is given, the position of the fold line can be obtained with the formula given in Section

2. Then the mechanical behavior of the cylinder during the motion was studied analytically. It can be found that

the structure has a bistable behavior. For the case with n=6, the bistable phenomenon can only occur when b/a

belongs to (1, 2). Finally, the numerical analysis was used to prove the accuracy of the analytical results.

Moreover, the folding process of the cylinder with multi-segments was also discussed.

In the present case, the bars are assumed to be elastic during the motion. The elasto-plastic behavior of the

cylinder will be investigated in future. Another problem is to deal with the structure with a shell element, which

is very complex and difficult issue.

Acknowledgement

The work presented in this article was supported by the National Natural Science Foundation of China (Grant

No. 51308106 and No. 51278116), the Natural Science Foundation of Jiangsu Province (Grant No.

BK20130614), the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No.




20130092120018) and a Project Funded by the Priority Academic Program Development of Jiangsu Higher

Education Institutions.

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the snap-back behavior of a self-deploying cylinder based on origami with kresling pattern

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