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Nonlinear dynamics of an adaptive origami-stent system
Accepted Manuscript
Nonlinear dynamics of an adaptive origami-stent system Guilherme V. Rodrigues , Larissa M. Fonseca , Marcelo A. Savi , Alberto Paiva PII: DOI: Reference:
S0020-7403(17)30887-1 10.1016/j.ijmecsci.2017.08.050 MS 3902
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
6 April 2017 20 August 2017 26 August 2017
Please cite this article as: Guilherme V. Rodrigues , Larissa M. Fonseca , Marcelo A. Savi , Alberto Paiva , Nonlinear dynamics of an adaptive origami-stent system, International Journal of Mechanical Sciences (2017), doi: 10.1016/j.ijmecsci.2017.08.050
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Highlights
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Dynamical analysis of origami systems. Geometrical and kinematic analysis of origami systems. Proposition of an archetypal single-degree of freedom model to describe origamistent dynamics. Numerical simulations considering different thermomechanical loadings that represent distinct operational conditions.
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NONLINEAR DYNAMICS OF AN ADAPTIVE ORIGAMI-STENT SYSTEM Guilherme V. Rodrigues 1 Larissa M. Fonseca 1 Marcelo A. Savi 1 Alberto Paiva 2 1
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Universidade Federal do Rio de Janeiro COPPE - Department of Mechanical Engineering Center for Nonlinear Mechanics 21.941.972 - Rio de Janeiro - RJ, Brazil E-Mails: [email protected]; [email protected]; [email protected] 2
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Universidade Federal Fluminense Volta Redonda School of Engineering Department of Mechanical Engineering 27.255.250 - Volta Redonda − RJ – Brazil E-Mail: [email protected]
ABSTRACT
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Origami is an ancient art of the paper folding that has been the source of inspiration in many engineering designs due to its intrinsic ability of changing shape and volume. Self-expandable
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devices have been created based on origami concept together with smart materials. Shape memory alloys belong to this class of materials and provide high forces and large displacements
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by varying their temperature. This work deals with the nonlinear dynamics of an origami-stent, a cylindrical shaped origami structure, which has the capacity of changing its radius. The actuation
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is provided by antagonistic torsional shape memory alloy wires placed in the origami creases. The mathematical model assumes a polynomial constitutive model to describe the shape memory alloy thermomechanical behavior. Geometric assumptions establish a one-degree of freedom
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model with constitutive and geometric nonlinearities. Numerical simulations are carried out considering different thermomechanical loadings that represent operational conditions. The system presents complex responses including chaos. Keywords: Origami; adaptive structures; shape memory alloys; nonlinear dynamics; chaos.
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1 - INTRODUCTION Origami, the art of the paper folding, has been inspiring many innovative designs of engineering systems. This Japanese art creates a three-dimensional geometry by following a sequence of folds in a sheet of paper. Some of these sequences creates a geometry able to contract-expand itself with an expressive change in its shape. Such behavior is of great interest to
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self-expanding adaptive systems.
The capacity of compacting structures, the lightness and the ability to present synchronized movements is of special interest of aerospace, robotic and medical applications (Fei and Sujan, 2012; Peraza-Hernandez, 2014). In aerospace applications, the need and transportation
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restrictions motivate the use of packed configurations and, once arrived at the desired place, be deployed to the expanded configuration. Miura (1994) developed a foldable solar panel based on origami concept. In medical applications, origami has inspired the design of new medical apparatus especially dedicated to minimally invasive surgery.
Shape memory alloys (SMAs) are widely employed as actuators due to their capacity of
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generate high forces and large displacements when subjected to thermomechanical fields that yield phase transformations. The association of SMA actuators with origami structures is
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promising being the focus of some research efforts (Koh and Cho, 2013; Pesenti et al., 2015; Kim et al., 2016). Lee et al. (2013) developed an origami-wheel actuated by SMA connected to a bias
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linear spring, being able to alter its radius in a robotic vehicle. The waterbomb pattern is employed for the origami design as employed on several applications since it allows significant shape changes (Hanna et al., 2014; Ma and You, 2014). Le et al. (2014) also used a waterbomb
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origami pattern in an aerial drone employed to landing in water. Salerno et al. (2014) presented an SMA origami device that takes advantage of geometric relations of the origami movements to
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increase the maneuver capacity. Origami system is a slender structure with complex geometry that makes stability analysis an
essential part of the design. Therefore, dynamical behavior of origami systems is an important issue to be investigated and only few works are available in literature. Basically, this analysis combines geometric and constitutive nonlinearities, providing a rich dynamics with operations close to the stability limits. In this regard, Fonseca et al. (2016) presented a dynamical analysis of
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an origami-wheel with shape memory actuators. A single degree of freedom system is employed as an archetypal model of the origami structure. Results show a rich response, including chaos. This work investigates the nonlinear dynamics of an origami-stent, a self-expandable cylindrical origami being originally presented by Kuribayashi et al. (2006). The stent is usually employed as a tubular medical device used to protect weakened arterial walls in the human body,
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but it can also be applied to increase stiffness of several systems including pipes and oil drilling well. Basically, origami-stent has a cylindrical shape and the folds allow the change of its radius
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(Figure 1).
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Figure 1 – Origami-stent deployment: three different configurations from folded, closed one, to
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deployed, opened one.
A shape memory alloy origami-stent system is analyzed considering torsion SMA wires
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(TSW) as actuators. A polynomial constitutive model describes thermomechanical behavior of SMAs. Origami geometric relations are employed to propose a one-degree of freedom (1-DOF)
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system that represents the system dynamics. Numerical simulations are carried out showing rich responses due to situations related to different thermomechanical operational excitations.
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After this introduction, this paper is organized as follows. Initially, a geometric analysis is
performed, establishing geometric relations and the kinematics of the origami system. Dynamical model is then proposed considering a single-degree of freedom system. Afterward, numerical simulations are discussed considering different thermal and thermomechanical loadings. Concluding remarks are presented in the sequence.
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2 - GEOMETRIC ANALYSIS The origami-stent is a self-expandable cylindrical origami constructed by following the folding pattern showed in Figure 2-a, where the continuous lines mean mountain folds (folds outwards the paper plane) and the dashed lines mean valley folds (folds inwards the paper plane). This pattern is composed by the repetition of rectangular components as showed in Figure 2-b,
side. The rectangular element has a length
and an angle
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which is known as waterbomb base, shifted half an element in comparison with those on their , and therefore,
⁄
and
means it is a square element. A single waterbomb base folded is presented in Figure 2-
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c.
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Figure 2 – Origami-stent folding pattern: a) waterbomb folding pattern where continuous line represents mountains and dashed line represents valleys; b) plane view of the single cell element;
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c) three-dimensional view of the single cell element.
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Geometric relations are now in focus and some hypotheses are established for this aim. The
first one is the symmetry, assuming that all elements behave in the same way and the element itself behaves symmetrically. Under this assumption, one-quarter of the cell element is representative of the general cell behavior. The second assumption is that fold occurs only in the creases and therefore, the element facets remain straight. Under these assumptions, it is possible to analyze the whole origami from a single cell element.
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The deployment of the origami goes through two stages (Kuribayashi, 2004). On the first stage, the element unfolds until it reaches a flattened configuration and the nodes A, C and A2 are
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co-linear. On the second stage, the crease AA2 moves outwards in the radial direction (Figure 3).
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Figure 3 – Origami-stent two-stage configurations: a) row of elements;
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b) single element.
The cell element geometry is characterized by three angles: ,
and
. These angles are
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coupled in such a way that each one can be described as a function of the others. Hence, it is possible to elect
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while
as the basic angle. It is noticeable that
characterizes the first stage
characterizes the second stage.
On the first stage,
can be expressed by considering the positions of the points A and B on
the (x,y,z) system with origin on C: A = (
) and B = (
).
Since the distance AB is equal to , the following equation can be written: (1)
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By solving Equation (1), the following equation is obtained: )
Based on that, there are two possible solutions for
(2) . Since this angle cannot be constant on
the first stage, the following equation is established to define ( In order to analyze the angle Since ApC =
(3)
)
(4)
, it is possible to write ApDp = and, since the angle ∠ADDp =
.
, the distance ApDp
. Therefore, the following relation is written for the first stage: ⁄
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(
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=
)
on the first stage, consider the triangle ApBpC (Figure 4).
and DpC =
On the other hand, AD =
during the first stage:
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(
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Figure 4 – Geometric characterization using angles of the origami-stent single element.
In order to obtain a function of
that depends only of , the angle
AC
substituted by the Equation (3) and it is found that On the second stage, Figure 3 shows that the angle
in the Equation (4) is .
remains constant, being equal to
.
This can be also pointed out by considering the second term on the left side of Equation (2). Besides, since both facets that contains the crease BC remains on the same plane,
.
An analysis of the origami-stent radii is now of concern. Let O be the central point of the origami-stent. Figure 5 shows two radii that define origami geometry: an internal radius,
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, defined as OBp. The angle , defined as ∠ApOBp, is
defined as OC; and an external radius,
related to the number of cell elements circumferentially distributed, n, and by the hypotheses already assumed, it is equal to all elements. Therefore, this angle is such that
.
Hence: (5) is the angle ∠ApCBp, it is possible to see from Figure 5 that
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Reminding that
, furnishing the following relations:
(6) (7)
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and
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Figure 5 – Geometric characterization using internal and external radii of the origami-stent:
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a) perspective view; b) frontal view.
The origami has physical limitations that can be expressed as follows: the radius
AC
less than zero, which means that the structure does not penetrate itself;
cannot be
cannot be greater than
, which means that the mountain AC does not become a valley avoiding a fold in another
direction that would result in the structure collapse. Hence, and
⁄
implies that
the definition of limit angles,
and
implies that
. These restrictions can be written by
: (8)
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(9) In summary, the origami geometry can be described with the following set of equations: {
(10) )
(11)
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(
{
Based on the geometric relations of the origami-stent, Figure 6 illustrates the general behavior
and
45°). Figure 6-a shows the evolution of the angles
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of an origami with square elements (
as a function of the angle . Note that at
, both angles ( and ) have a transition
configuration between the different stages. Figure 6-b shows internal and external radii divided by the length L for different values of n (the number of elements circumferentially distributed). Besides the limit angles,
and
, the number n influences the radius maximum values. occurs at
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The maximum value of the external radius
degree of compactness of the structure. For n = 6,
. Figure 6-b also shows the
can double its size, while for n = 10,
can
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triplicate its size. Hence, the greater is the value of elements n, the more compact the origami is able to be. Nevertheless, the increase of the number of elements makes harder the origami
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manufacturing process.
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90
60
30
0 30
45
60
()
75
90
Re
n = 10
Ri
n=8
2
n=6
1
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Radius / L
Angle (°)
3
0
105
15
30
45
60
75
90 105 120
Figure 6 – Geometric relations defined by origami angles and radius as a function of degrees for
= 45°): a) angle relations ( and ); b) radius relations (
) for different
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values of n.
and
(in
2.1 - Kinematic Analysis
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Kinematics of the origami-stent system is now treated. Due to symmetry assumptions, it is possible to analyze one-quarter of the cell element. A global coordinate frame F(OXYZ) is placed
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on the center axis of the origami-stent with origin on O (see Figure 5). A local coordinate frame P(Cxyz) is placed due to a translation from O to C (see Figure 5). Hence, the coordinate frame P is translated from the coordinate frame F through Z direction of a distance equal to OC, that is,
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.
Basically, kinematic analysis corresponds to two triangles which movements are
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geometrically coupled to each other. It is assumed that each one of these triangles rotates presenting inertia. The triangle BBpC rotates of an angle and then of an angle
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angle
(see Figure 4). Besides that, translation motion associated with
each triangle has a lumped mass placed in its centroid: and
while the triangle ABC rotates of an
on the centroid of the triangle ABC
on the centroid of the triangle BBpC.
Therefore, five coordinate frames are considered: F and P; Q, R and S. The coordinate frames Q, R and S have origin on C, presenting movements related to each geometric angle, , respectively (Figure 7). The triangle ABC rotates about
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and ,
and, then, rotates about
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Q(Cx(1)y(1)z(1)) rotates about
in the x(1)-axis direction of P while R(Cx(2)y(2)z(2)) rotates about
in the z(2)-axis direction of Q. The triangle BBpC rotates . S(Cx(3)y(3)z(3)) rotates about
on the
y(3)-axis of P. Since the coordinate frame P does not rotates about the global coordinate frame F (it only translates), the rotation matrices are the following: [
]
[
]
(13)
[
]
(14)
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(12)
Figure 7 – Coordinate frames defined on the origami-stent single element.
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It is assumed that the lumped masses
and
are located in the geometric center of the
triangles ABC and BBpC, respectively. The vectors that indicate their positions are represented on the coordinate frames R and S, respectively, given by: ]
[
]
(15)
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[
(16)
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The triangle inertia tensors are the following:
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[
]
(18) ]
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[
(17)
After these definitions, it is defined triangle angular velocities with respect to the global
̇
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The lumped mass linear velocities
̇+ ̇
*
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frame:
(19)
[ ̇]
and
are given by the position time derivative,
reminding that all angles are described as a function of
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(20)
.
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[
] ̇ ̇ ̇
[
(
(21)
* ̇
(
* ̇
̇
]
[
]
[
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̇
]
̇ 3 - DYNAMICAL MODEL
(22)
̇
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The dynamical analysis of the origami-stent system assumes kinematic relations in order to write an equivalent 1-DOF system to represent the origami dynamics. Equations of motion are formulated by considering energetic approach, electing angle
as a generalized coordinate.
Using the transformation matrices to describe all terms in the system F, the kinetic energy
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given by:
(23)
)
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(
Figure 8 shows the kinetic energy E as a function of
is
and its time derivative, ̇ , for a square
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element, considering n = 6, unitary length L and unitary masses. Figure 9 shows the projection of this energy for different values of n, in the left as a function of ̇ for for a unitary velocity ( ̇
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function of
1°/s). It should be pointed out that the energy is a
90° due to changes of the angles
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nonsmooth function at
= 90° and in the right as
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and
(see Figure 6).
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Figure 8 – Kinetic energy of origami-stent square element with unitary length and unitary masses
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5
0 -1.0
0.0
0.5
1.0
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-0.5
E (10-3 J)
10
n=6 n=8 n = 10
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E (10-3 J)
10
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and six elements (n = 6).
n=6 n=8 n = 10
5
0
15 30 45 60 75 90 105
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Figure 9 – Projections of the kinetic energy of origami-stent square element with unitary length = 90°; b) ̇
1 °/s.
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and unitary masses for different number of elements, n; a)
The actuation of the origami-stent is provided by a Torsion SMA Wire (TSW) with length
and radius
(Koh et al., 2014). Two TSWs are antagonistically placed on each side of a cell
element on the crease AC (Figure 10) and these actuators promote the origami configuration changes (opening and closing).
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Figure 10 – Actuation system provided by Torsion Spring Wires (TSWs): a) the placement of the actuators at the planar single element; b) a folded single element with TSWs and lumped masses
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placed on the triangles’ centroids; b) Torsional Spring Wire (TSW).
The SMA thermomechanical behavior is described by a polynomial constitutive model (Falk, 1980; Savi & Pacheco, 2002) that establishes a stress-strain-temperature
relation.
This constitutive model considers that the austenitic phase is stable at high temperatures, and two variants of the martensitic phase, induced by positive and negative stress fields, stable at low
,
and
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where
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temperatures.
(24)
are material parameters and
as the temperature above which the austenitic
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martensitic phase is stable. By considering
is the temperature below which the
phase is stable, it is possible to write the following relationship:
.
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Aguiar et al. (2010) and Enemark et al. (2016) showed that helical spring force-displacementtemperature curve is similar to the stress-strain-temperature curve assuming homogeneous phase
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transformation through the spring element. Under this assumption, it is possible to use the same argues to define an torsional actuator torque-displacement-temperature relation that is formally similar to the stress-strain-temperature: * (
( )
) (
15
) ]
(25)
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where
is the angle where the TSW is free of stress. Note that this expression represents
constitutive relation of both actuators, being expressed by
and
On the origami-stent system, the generalized forces
are related to the TSW generalized
, which due to symmetric considerations is given by:
An external generalized force the virtual work
.
is applied at the point C, z-direction. Hence, the expression of
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force
.
, neglecting gravitational effect, is given by:
(26)
Note that
⁄
,
, and therefore,
⁄
.
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Two extra generalized forces are considered in order to have a complete description of the origami-stent system. The first one is related to a linear viscous dissipation of coefficient ̇
representing all dissipation processes, including the hysteresis phenomenon: second one is related to origami geometric restrictions that limit
. The
to vary in the range [θmin, θmax]
(27)
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{
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(Equations (8) and (9)), expressed as a moment of the form:
where klim limits the system response and represents an equivalent elastic spring stiffness.
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Equation of motion is then obtained from Lagrange’s equation. ̇
( )
(28)
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( ̇)
The kinetic energy (23) is a nonlinear expression in
multiplied by the velocity and can be
̇ . Details about this function are presented in Appendix. Substituting this
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written as
expression in the left hand side of Lagrange’s equation (28), it gives a term
̈ and a term
̇ (see Appendix for more details). Therefore, the equation of motion is written as follows: ̈
̇
̇
(29)
In order to write dimensionless equations, it is defined a dimensionless time where
is a reference frequency, defined from the linear part of the restitution force:
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,
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√
(30)
Hence, after dividing the Equation (29) by
, the following set of equations of
motion is written: { ⁄
where
, and therefore
̇
̈
and
dimensionless terms are employed:
)
(31)
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(
. Besides, the following
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(32)
(34) (35) (36)
Figure 11 presents dimensionless functions
and at
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masses. The discontinuity of the function
(33)
for a square element and equal
90° is due to the nonsmoothness of the
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kinetic energy. These functions indicate the origami-stent nonlinearities.
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1.0
0.15
0.10
f2
0.0
0.05
n=6 n=8 n = 10 -1.0 15 30 45 60 75 90 105 -0.5
Figure 11 – Dimensionless functions
0.00
and
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f1
0.5
n=6 n=8 n = 10
15 30 45 60 75 90 105
that define geometrical nonlinearities of = 45°).
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the dynamical model (
4 - NUMERICAL SIMULATIONS
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Dynamical behavior of the origami-stent system actuated by two antagonistic TSWs is now analyzed. Numerical simulations are carried out by employing the fourth-order Runge-Kutta
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method. Different thermal and thermomechanical loads are contemplated to study the system behavior under distinct operational conditions.
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Table 1 presents the origami-stent parameters employed in all simulations considering a square element. Besides, a dissipation coefficient
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the reference frequency is
= 0.3 is employed. Under these conditions,
rad/s. Temperature
represents a reference
temperature where the origami-stent actuator does not apply any moment at a chosen angle of
AC
(closed configuration). Thereby, one actuator has
. The other one has
since this assures that the origami can reach the opened configuration of maximum radius (
) by changing the TSW temperature. This set of parameters can represent a human body
stent. Note that the length =13.2 mm for
of the cell element is 6.6 mm. Hence, for n = 6, the external radius
= 90° (equation (6)), which is close to the usual radius of aortal stent grafts.
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Figure 12 presents the torque of the TSWs as a function of 310K (
). TSW-1 is located at
with a temperature
at the temperature
with a temperature
and at T =
while TSW-2 is at
.
(kg)
(kg) 510-5
510-5
(MPa/K)
(N m)
8.010-5
(MPa)
(MPa)
(K)
(K)
(K)
1.41010
2.21012
287.15
309.1
293.15
2.210-3
1.010-2
0.5
M
0.0 TSW-1 TSW-2 0
45
PT
-1.0 -90 -45
90
135 180
M (10-6 N m)
1.0
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M (10-6 N m)
(m)
6.610-3
1.0
-0.5
(m)
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1.0106
(m)
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Table 1 – Origami-stent parameters.
0.5 0.0
-0.5
TSW-1 TSW-2
-1.0 -90 -45
0
45
90
135 180
Figure 12 – Torque of the TSWs at different temperatures: a) reference temperature
;
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b) T = 310 K.
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Concerning external excitation, a harmonic external force is considered of amplitude
and
frequency .
(37)
In addition, temperature has fluctuations represented by: (38)
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where
is the temperature around which the temperature
oscillates.
4.1 - Thermal Loadings The actuator temperature influence on the system dynamical response is investigated considering different thermal loadings. An overview of the origami-stent equilibrium
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configurations can be provided by the analysis of the moments represented by (Equation (34)). Figure 13 presents these moments for different pairs of temperatures and n = 6. In the reference configuration (Figure 13-a),
=
three angles: 30°, 60° and 90°. If the temperature
=
, the sum of the moments vanishes at = 310 K) while
=
, only 30° is an equilibrium point (Figure 13-b). On the other hand, if the temperature
is
(
= 310 K) while
=
(
, only 90° is an equilibrium point (Figure 13-c). If the
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greater than
is greater than
temperatures of both TSWs are greater than
(
=
= 310 K), there are again three points
where the sum of the moments is zero (Figure 13-d). Therefore, by controlling the temperature one can establish the configuration (opened or closed) of the origami-stent, defining the
AC
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PT
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equilibrium point structure.
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0.08
0.04
0.04
H1
H1
0.08
0.00
-0.04 -0.08
-0.04 30
45
60
75
90
-0.08
105
0.08
45
60
75
90
105
60
75
90
105
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0.08
0.04
H1
H1
0.04 0.00
0.00
-0.04
30
45
60
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-0.04 -0.08
30
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0.00
75
90
105
-0.08
30
45
ED
Figure 13 – Actuator temperature influence on the generalized forces: a) TSW-1 and TSW-2 at reference temperature with three vanishing points; b) TSW-1 at high temperature with one
PT
vanishing point at 30°; c) TSW-2 at high temperature with one vanishing point at 90°; d) TSW-1
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and TSW-2 at high temperature with three vanishing points.
Free vibration analysis is now of concern identifying the temperature influence. Figures 14
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and 15 present the dynamical behavior of the system for different initial conditions and temperatures. The temperature of one actuator is modified while the other remains constant at the reference temperature. Figure 14 presents the phase space at different temperatures for the TSW1 while TSW-2 temperature is at
=
. The sum of the moments points up to three equilibrium points
, and the phase space shows that two of them are stable while the other one is unstable.
By increasing the temperature
(
= 306 K), 30° remains a stable equilibrium point while the
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other stable equilibrium point is, now, smaller than 90°. When
>
(
= 310 K), only 30° is
an equilibrium point, meaning that the structure admits only the closed configuration. 5.0
0.08
H1
0.04 0.00
-0.04 5.0 30
45
60
75
90
v (degree) v (degree)
30
30
45
45
60
75
(degree) 60
75
(degree)
90
105
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-2.5 -5.0
90
60
105
75
(degree)
60
75
(degree)
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0.0 -2.5
45
45
5.0 2.5
2.5 0.0
-5.0
30
-4 5.0 30
105
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v (degree) v (degree)
5.0 2.5
0 -2.5 -2 -5.0
T1 = T 0 T1 = 306 K T1 = 310 K
-0.08
2 0.0
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v (degree) v (degree)
4 2.5
90
90
105
105
2.5 0.0
0.0 -2.5
-2.5 -5.0 -5.0
30
30
45
60
75
(degree)
45
60
75
(degree)
90
90
105
105
Figure 14 – Origami free vibration response at different temperatures with 6 elements (n = 6). and different values of
PT
a) Sum of the moments at
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space at
. b) Phase space at
. d) Phase space at
. c) Phase
.
Figure 15 presents the phase space at different temperatures for the TSW-2 while the . By increasing the temperature
(
= 306 K), 90° remains a
AC
temperature of the TSW-1 is
stable equilibrium point while the other stable equilibrium point is now, greater than 30°. When >
(
= 310 K), only 90° is an equilibrium point, and the structure only admits an opened
configuration. It is important to highlight that the origami-stent is initially in a reference configuration where the structure is closed ( = 30°). The opened configuration can be reached either by temperature variation or by imposing proper initial conditions, as one can see in Figure
22
ACCEPTED MANUSCRIPT
15-a (or Figure 14-a). This reveals an initial condition dependence that is now investigated using different thermal loads.
5.0
0.08
H1
0.04 0.00
-0.04 45
60
75
90
v (degree) v (degree)
4 2.5
45
30
45
60
75
(degree)
90
ED
30
M
0 -2.5
-4
-4 5.030
105
45
45
60
75
90
(degree)
60
75
90
(degree)
105
105
4 2.5
2 0.0
-2 -5.0
30
AN US
30
0 -2.5 -2 -5.0
T2 = T 0 T2 = 306 K T2 = 310 K
60
75
(degree)
v (degree) v (degree)
-0.08 5.0
2 0.0
CR IP T
v (degree) v (degree)
4 2.5
105
90
105
20.0
0 -2.5
-2 -5.0 -4
30
30
45
45
60
75
90
(degree) 60
75
(degree)
90
105
105
PT
Figure 15 – Origami free vibration response at different temperatures with 6 elements (n = 6). a) Sum of the moments at
CE
space at
and different values of
. d) Phase space at
From the reference condition, when temperature
AC
. b) Phase space at .
increases up to 310 K, the origami-stent
opens (Figure 16). System response presents an oscillation before reaching temperature
decreases to
increasing temperature By decreasing
to
. c) Phase
= 90°. If the
again, the origami-stent remains in the opened configuration. By
, 90° is not an equilibrium point anymore and the origami-stent closes.
, the structure remains closed.
23
ACCEPTED MANUSCRIPT
315
105
T2
310
90
305
75
(°)
300
45
295 290
60
30 0
2000
4000
6000
8000
0
CR IP T
(K)
T1
2000
4000
6000
8000
Figure 16 – Opening-closing process induced by temperature variations (n = 6): a) thermal
AN US
load; b) time history of the deployment angle θ.
A different thermal loading is now investigated. By increasing subsequently increasing
up to 310 K and
up to 310 K (Figure 17), the structure remains opened but not
completely, since the equilibrium point is around 82°. By decreasing the temperature there is no more changes.
T1
305 300
CE
295
0
AC
290
90
PT
(K)
310
105
T2
(°)
315
ED
M
origami-stent closes and by decreasing
, the
2000
75 60 45 30
4000
6000
0
8000
2000
4000
6000
8000
Figure 17 – Opening-closing process induced by temperature variations (n = 6): a) thermal load; b) time history of the deployment angle θ.
24
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is first increased (Figure 18), there is no structure change. By increasing
subsequently, there is a small increase of the angle the stent-origami opens and, subsequently, if
90
305
75
(°)
(K)
105
T2
310
300
60 45
295 290
30 0
2000
decreases,
decreases, it remains opened.
315
T1
and it stabilizes around 38°. If
4000
6000
0
2000
4000
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8000
CR IP T
In contrast, if
6000
8000
Figure 18 – Opening process induced by temperature variations (n = 6): a) thermal load; b) time
M
history of the deployment angle θ.
Equilibrium configuration small increase can be obtained increasing both temperatures
ED
simultaneously as presented in Figure 19. Note that the angle 315
T1
90
300
(°)
PT
305
AC 0
75 60 45
295 290
is changed to 38°.
105
T2
CE
(K)
310
and
30
1000
2000
3000
0
1000
2000
3000
Figure 19 – Opening process induced by temperature variations (n = 6): a) thermal load; b) time history of the deployment angle θ.
25
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Until now, all simulations consider the same number of elements, n = 6. The change of the number of elements can influence the dynamical response, which is now investigated. The first point that should be observed is that when n changes, the limit angles change; both tend to reduce for greater values of n. For n = 6,
= 30° and
and
= 105°. For n > 6,
< 30° and the origami is not completely closed at the reference condition and
< 105°.
Figure 20 presents the origami response when temperature
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Besides that, Figure 9 shows that the amount of kinetic energy is larger for greatest values of n. increases from the reference
condition causing the opening of the origami. Different values of n are treated. Although the actuation response occurs at the same time, the stabilization takes more time to occur for origami with larger number of elements due to the greater amount of kinetic energy.
T1
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315
105
T2
90
(°)
305 300
290
75 60 45
295
0
1000
2000
3000
M
(K)
310
4000
n=6
30
5000
0
1000
2000
90
(°)
60 45
AC
1000
75 60 45
n=8
30
0
5000
105
PT
75
CE
(°)
90
4000
ED
105
3000
2000
3000
4000
n = 10
30
5000
0
1000
2000
3000
4000
5000
Figure 20 – Comparison of the opening process considering different values of the number of elements, n. a) thermal load; time history of the deployment angle θ for: b) n = 6; c) n = 8; d) n = 10.
26
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Since the origami system has strong temperature dependent behavior, it is important to evaluate the effect of thermal oscillations on system response. Figures 21 to 23 present the influence of thermal oscillations on temperatures
. Afterward, both temperatures oscillate at the
opened configuration. A variation of temperature the temperature
considering n = 6. Figure 21 presents
does not alter the response, but a variation of
CR IP T
the opening process changing temperature
and
causes small oscillations. The TSW-2 is at a high temperature and 90° is a
free-stress point (see Figure 12) that does not change with the thermal variation. Based on that, the sum of system moments is not altered and, therefore, its equilibrium point does not change. However, the TSW-1 is at the reference temperature and thermal fluctuation alters the free-stress
oscillates while the temperature
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points, except the one at 30°. Hence, the sum of the moments is altered and its equilibrium point oscillates.
315 T1
105
T2
M
305 300 295 290
0
1000
2000
3000
4000
(°)
90
ED
(K)
310
75 60 45 30
5000
0
1000
2000
3000
4000
Figure 21 – Opening process with thermal oscillation (n = 6): a) thermal load (
= 2.0 K and
= 0.1); b) time history of the deployment angle θ.
AC
CE
5000
PT
Figure 22 shows a similar opening condition but with a situation where both
increased. Under this assumption, a variation of temperature
and
are
also induces oscillations and the
origami response is a combination of both influences. Since both actuators have
, a
variation of any actuator temperature is responsible to alter the sum of the moments of the system and, as a consequence, there is an oscillatory response. Besides, frequency thermal frequency
27
ACCEPTED MANUSCRIPT
phase is analyzed. It is noticeable that when actuator thermal variation is out of phase, the oscillatory response has higher amplitude.
315
90
305
75
310
300
60
295
45
290
30 0
1500
3000
4500
315 T1
6000
0
1500
3000
4500
6000
3000
4500
6000
105
T2
310
90
305
75 60
300
M
(K)
CR IP T
105
T2
AN US
(K)
T1
295 0
1500
3000
4500
ED
290
6000
45 30 0
1500
Figure 22 – Opening process with thermal oscillation (n = 6): a) and c) thermal load (
= 2.0
PT
= 0.1); b) and d) time history of the deployment angle θ.
CE
K and
Figure 23 presents a situation where both actuators have
AC
configuration. Under this condition, only the variation of contrast of the case of Figure 21, small fluctuation of
, characterizing a closed affects the dynamic response. In
does not alter actuator dynamical
response since 30° is an actuator free-stress point (see Figure 12) that does not change with a thermal variation. On the other hand, TSW-2 has 90° as a free-stress point that does not change, while the other points change with the temperature and, hence, thermal fluctuation changes the sum of the moments, affecting the system equilibrium point.
28
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315
105
T2
310
90
305
75
(°)
300
45
295 290
60
30 0
500
1000
1500
2000
2500
0
500
CR IP T
(K)
T1
1000
1500
2000
Figure 23 – Closed configuration with thermal oscillation (n = 6). a) Thermal load (
= 2.0 K
= 0.1); b) time history of the deployment angle θ.
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and
2500
The influence of the number of elements in a case with thermal fluctuation is presented in Figure 24. Note that the increase of the number of elements, n, tends to cause a greater change on
AC
CE
PT
ED
M
system response.
29
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315
105
T2
310
90
305
75
(°)
300
45
295 290
60
n=6
30 0
1500
3000
4500
6000
0
1500
3000
4500
6000
105
90
90
75
75
(°)
105
60 45
AN US
(°)
CR IP T
(K)
T1
60 45
n=8
30 0
1500
3000
4500
0
1500
3000
4500
6000
M
6000
n = 10
30
Figure 24 - Comparison of the opening process with thermal oscillation considering different for:
ED
values of the number of elements, n. a) Thermal load; time history of the deployment angle
PT
b) n = 6: c) n = 8; d) n = 10.
CE
4.2 - Thermomechanical Loadings Thermomechanical loadings are now addressed in order to simulate some operational =
AC
conditions of the origami-stent. Basically, opened configuration is of concern considering and
= 310 K. Under this condition, only 90° is an equilibrium point and an external force is
applied. In order to establish a physical interpretation, the origami-stent system is assumed to represent human body stent. Under this assumption, the magnitude of the external force is compatible to the human body blood pressure. Hence, the dimensionless amplitude external force
is in a range from 0.01 to 0.85, which is related to force of magnitude
30
of the N.
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Initially, a global analysis is performed considering bifurcation diagrams that present a stroboscopic view of system dynamics under the slow quasi-static variation of parameter
for
0.30 and different number of elements: n = 6; n = 8; n = 10 (Figure 25). Figure 25-a considers n = 6. Note that, for small force amplitudes, the system presents period-1 response until = 0.143. After that, a cloud of points suggests a chaotic response. By increasing the value of ,
When
CR IP T
the diagram shows other regions where the system response has a periodicity greater than 1. = 0.156, bifurcations start from chaotic-like response to period-1 response. A period
doubling can be noticed at
= 0.260, nevertheless a period-1 response is predominant until
=
0.628, where the system response becomes chaotic. Inside this chaotic region, there are some periodic windows, especially between
= 0.740 and
= 0.781.
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By considering different number of elements, n = 8 (Figure 25-b) and n = 10 (Figure 25-c), it is noticeable that the increase of the number of elements tends to stretch the diagram and spread the kind of behavior over the values of . Note, for instance, that period-1 response for small amplitudes of the force finishes at
= 0.143 when n = 6 but at
= 0.157 when n = 8 and at
=
0.308 when n = 10. After the region of period-1 response, different bifurcation evolutions occur
M
for each cell number. When n = 6, the cloud of points occurred immediately after the period-1 response. When n = 8, a period doubling takes place until the system response is quasi-periodic
ED
and then, the cloud of points takes place. When n = 10, a period doubling occurs then the cloud of points takes place. Subsequently, a periodic response window is noticed until there is a response
PT
sudden change suggesting a chaotic response. Afterward, a bifurcation from chaos to period-1
AC
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occurs.
31
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(degree)
105 75 45
0.0
0.2
0.4
0.6
75
75
15 n = 8
45 15
0.2
0.4
0.6
0.8
M
0.0
0.8
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(degree)
105
(degree)
105
45
CR IP T
15 n = 6
n = 10
0.0
Figure 25 - Bifurcation diagrams varying excitation amplitude
0.4
0.6
0.8
for different number of elements
0.30): a) n = 6; b) n= 8; c) n = 10.
ED
and a fixed frequency (
0.2
PT
Phase space and its respective Poincaré section are presented in Figure 26 for different amplitudes of the external force taken from the bifurcation diagram of n = 6. In all these cases,
CE
the external force is great enough to push the origami to its physical limits, which is indicated by the straight lines at 30° and 105°. For
= 0.145, a point inside the cloud of points region, a
AC
chaotic response occurs. By increasing values of , period-1 and period-2 responses are achieved for
= 0.20 and
= 0.36, respectively. A chaotic response is again obtained for
32
= 0.70.
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5.0
5.0 16
16
2.5
(degree) vv(°)
0.0 0
-8
-5.0 30 -16 15 30 5.0
-5.0
45 45
60 75 90 105 60(degree) 75 90 105 120
(°)
45 30
45
v (°) v (degree)
30 2.5
10 0.0 0 -2.5
15 0.0
60
75
(degree) 60
75
(°)
AN US
v (°) v (degree)
30
-16 15 5.0
20 2.5
90
90
105
105
120
0 -2.5
-15
30 30
45 45
60
75
90
(degree) 60
75
(°)
90
105
105
-5.0 30 -30
120
15
M
-20 15
0
-2.5
-2.5 -8
-10 -5.0
8
0.0
CR IP T
vv (degree) (°)
2.5 8
Figure 26 – Origami-stent response for
30
45
60
75
90
(degree) 60
75
(°)
90
105 105
120
= 0.3 and n = 6. Phase space (black lines) and
= 0.145. b)
= 0.200; c)
= 0.360. d)
= 0.700.
ED
Poincaré section (red points). a)
45
PT
The same results of Figure 26, that considers n = 6, is now presented for n = 8, Figure 27, and n = 10, Figure 28. When
= 0.145 (Figures 27-a and 28-a), the system oscillates around 90° with
CE
a period-1 response, without reaching the physical limits of the structure. When
= 0.20, there is
a quasi-periodic response for n = 8 and, even if the system still oscillates around 90°, the upper
AC
physical limit is reached. In contrast, for n = 10, this increase of response substantially. On the other hand, when 8, but chaotic for n = 10. When
does not affect the system
= 0.36, the system response is periodic for n =
= 0.70, a period-1 response is noticed.
33
5.0
5.0
4 2.5
4 2.5
2 0.0 0 -2.5 -2 -5.0
30
45
-4 60 5.0
60
75
90
(degree) 75
(°)
v (°) v (degree)
0 -2.5
30 30
45 45
60
75
90
(degree) 60
75
(°)
90
105
105
Figure 27 – Origami-stent response for
120
75
90
(°)
90
105 105
0 -2.5
-16 15
30
30
45
45
60
75
90
(degree)
60
75
(°)
90
105
105 120
= 0.3 and n = 8. Phase space (black lines) and
= 0.145. b)
AC
CE
PT
ED
Poincaré section (red points). a)
8 0.0
-8 -5.0
M
-16 15
60
(degree)
75
16 2.5
8 0.0
45
AN US
v (°) v (degree)
30
-4 60 5.0
105
16 2.5
-8 -5.0
0 -2.5 -2 -5.0
105
90
2 0.0
CR IP T
v (degree) v (°)
v (degree) v (°)
ACCEPTED MANUSCRIPT
34
= 0.200; c)
= 0.360. d)
= 0.700.
5.0
5.0
4 2.5
4 2.5
2 0.0 0 -2.5 -2 -5.0
30
45
-4 60 5.0
75
60
75
90
(degree) 90
(°)
105
30
45
-4 60 5.0
105
75
v (°) v (degree)
20 2.5
10 0.0 0 -2.5
10 0.0
30
45 45
75
90
60
75
(°)
90
105 105
120
M
30
60
(degree)
Figure 28 – Origami-stent response for
90
105
90
(°)
105
-20 15
30
45
30
45
60
75
90
(degree) 60
75
(°)
90
105 105
120
= 0.3 and n = 10 elements. Phase space (black lines)
= 0.145. b)
= 0.200; c)
= 0.360. d)
= 0.700.
ED
and Poincaré section (red points). a)
75
0 -2.5
-10 -5.0
-20 15
60
(degree)
AN US
v v(°)(degree)
0 -2.5 -2 -5.0
20 2.5
-10 -5.0
2 0.0
CR IP T
v v(°)(degree)
v v(°)(degree)
ACCEPTED MANUSCRIPT
PT
Resonant condition is usually critical to system response. Concerning chaotic behavior, they can be induced more effectively close to this condition. In this regard, Figure 29 presents the
CE
maximum amplitude of displacement under the variation of frequency . For n = 6 (Figure 29-a), it presents a typical resonant curve with maximum amplitude at
0.085, related to constitutive nonlinearities. For n = 8 and n = 10, the peaks of
AC
small peak at
0.159. It is also noticeable a
maximum angle is shifted to the left and the flat region related to the maximum values is due to the maximum displacement constraints imposed by the origami physical limits. Besides, the smaller peak is more pronounceable. Furthermore, Figure 29-d presents a zoom in Figure 29-c where dynamical jumps for different frequencies are observed.
35
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105
105
n=6
100
max
95 90 0.01
95 90
0.06
0.11
0.16
0.21
0.01
CR IP T
max
100
n=8
0.06
100.0
105
n = 10
max
95 90 0.01
0.06
0.11
97.5
AN US
max
100
0.11
0.16
0.21
M
Figure 29 – Origami-stent frequency diagram for
95.0 0.085
0.090
0.095
0.16
0.100
0.21
0.105
= 0.01 and different number of elements: a) n
= 6; b) n = 8; c) n = 10; d) enlargement of the resonance region for n = 10 elements showing
PT
ED
dynamical jumps.
Figure 30 shows the bifurcation diagram for n = 6 varying
under resonant condition (
CE
0.16). Note that system response is predominantly chaotic indicating the influence of the resonant condition on system dynamics. Phase space and Poincaré section is presented in Figure 31 for 0.16 and
= 0.20. A chaotic-like response is observed presenting a strange attractor in
AC
Poincaré section.
36
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75 45 15 n = 6 0.0
0.2
0.4
0.6
CR IP T
(degree)
105
0.8
20
5
v
0
M
v
0.16 and for n = 6.
10
10
30
45
ED
-10 -20
for
AN US
Figure 30 – Bifurcation diagram varying the excitation amplitude
60
75
-5 -10 30
105 ,
45
60
75
90
0.16 and n = 6. a) Phase space. b) Poincaré
section.
CE
PT
Figure 31 – Chaotic response for
90
0
AC
5 - CONCLUSIONS This paper deals with the nonlinear dynamical analysis of an origami-stent actuated by
shape memory alloys. A polynomial constitutive model is employed to describe the thermomechanical behavior of SMA actuators. Geometrical relations allow one to build a single degree of freedom system to analyze the system dynamics. This system has both constitutive and geometrical nonlinearities, presenting nonsmooth characteristics. Numerical simulations are carried out showing different thermomechanical loading situations representing operational 37
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conditions. Concerning thermal loadings, it is possible to identify that different configurations can be reached by changing the temperature of each actuator. Results are affected by the order of actuation. Moreover, thermal fluctuations can affect the system response depending on conditions. It depends on either the origami configuration or the actuator temperature. The number of cell elements interferes on the response stabilization time as well. An origami with
CR IP T
more cell elements takes more time to stabilize, being more affected by thermal fluctuations. Thermomechanical loadings can produce complex responses including chaos. The number of cell elements is critical in order to define the system response. In general, the increase of the number of cell elements reduces chaotic regions. Furthermore, greater values of external force are needed to push the origami to reach its physical limits. Dynamical jumps are also observed on origami-
6 - ACKNOWLEDGEMENTS
AN US
stent dynamics.
The authors would like to acknowledge the support of the Brazilian Research Agencies
M
CNPq, CAPES and FAPERJ. The Air Force Office of Scientific Research (AFOSR) is also
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7 - REFERENCES
ED
acknowledged.
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CE
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CR IP T
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Origami Enabled, SMA Actuated, Robotic End-Effector for Minimally Invasive Surgery. Proceedings of the IEEE International Conference on Robotics and Automation (ICRA2014), Hong Kong, pp. 2844–2849.
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Savi, M. A. & Pacheco, P. M. L. C. (2002), “Chaos and hyperchaos in shape
[19]
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memory systems”, International Journal of Bifurcation and Chaos, v.12, n.3, pp.645-657.
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APPENDIX Origami-stent equations of motion are related to complex functions that express geometric relations. Kinetic energy definition (Equation (23)) is described on the global coordinate frame F, using transformation matrix. Therefore, consider that kinetic energy, Equation (22), can be
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written as follows: ̇
Applying this equation in the left hand side of the Lagrange’s equation (Equation (28)):
̇
*
̇
( ̇
̈
*
̇
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(
̇ )
(
M ̈
*
̇
and
⁄
, reminding that
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the sequence, where
41
and
̈ ⁄
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Therefore,
̇
̇
̇
Combining the terms with ̈ and ̇ , two others functions, (
̈
and
are defined: ̇
. The equation are functions of :
is presented in
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Graphical Abstract
45
Nonlinear dynamics of an adaptive origami-stent system Guilherme V. Rodrigues , Larissa M. Fonseca , Marcelo A. Savi , Alberto Paiva PII: DOI: Reference:
S0020-7403(17)30887-1 10.1016/j.ijmecsci.2017.08.050 MS 3902
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
6 April 2017 20 August 2017 26 August 2017
Please cite this article as: Guilherme V. Rodrigues , Larissa M. Fonseca , Marcelo A. Savi , Alberto Paiva , Nonlinear dynamics of an adaptive origami-stent system, International Journal of Mechanical Sciences (2017), doi: 10.1016/j.ijmecsci.2017.08.050
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Highlights
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Dynamical analysis of origami systems. Geometrical and kinematic analysis of origami systems. Proposition of an archetypal single-degree of freedom model to describe origamistent dynamics. Numerical simulations considering different thermomechanical loadings that represent distinct operational conditions.
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NONLINEAR DYNAMICS OF AN ADAPTIVE ORIGAMI-STENT SYSTEM Guilherme V. Rodrigues 1 Larissa M. Fonseca 1 Marcelo A. Savi 1 Alberto Paiva 2 1
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Universidade Federal do Rio de Janeiro COPPE - Department of Mechanical Engineering Center for Nonlinear Mechanics 21.941.972 - Rio de Janeiro - RJ, Brazil E-Mails: [email protected]; [email protected]; [email protected] 2
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Universidade Federal Fluminense Volta Redonda School of Engineering Department of Mechanical Engineering 27.255.250 - Volta Redonda − RJ – Brazil E-Mail: [email protected]
ABSTRACT
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Origami is an ancient art of the paper folding that has been the source of inspiration in many engineering designs due to its intrinsic ability of changing shape and volume. Self-expandable
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devices have been created based on origami concept together with smart materials. Shape memory alloys belong to this class of materials and provide high forces and large displacements
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by varying their temperature. This work deals with the nonlinear dynamics of an origami-stent, a cylindrical shaped origami structure, which has the capacity of changing its radius. The actuation
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is provided by antagonistic torsional shape memory alloy wires placed in the origami creases. The mathematical model assumes a polynomial constitutive model to describe the shape memory alloy thermomechanical behavior. Geometric assumptions establish a one-degree of freedom
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model with constitutive and geometric nonlinearities. Numerical simulations are carried out considering different thermomechanical loadings that represent operational conditions. The system presents complex responses including chaos. Keywords: Origami; adaptive structures; shape memory alloys; nonlinear dynamics; chaos.
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1 - INTRODUCTION Origami, the art of the paper folding, has been inspiring many innovative designs of engineering systems. This Japanese art creates a three-dimensional geometry by following a sequence of folds in a sheet of paper. Some of these sequences creates a geometry able to contract-expand itself with an expressive change in its shape. Such behavior is of great interest to
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self-expanding adaptive systems.
The capacity of compacting structures, the lightness and the ability to present synchronized movements is of special interest of aerospace, robotic and medical applications (Fei and Sujan, 2012; Peraza-Hernandez, 2014). In aerospace applications, the need and transportation
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restrictions motivate the use of packed configurations and, once arrived at the desired place, be deployed to the expanded configuration. Miura (1994) developed a foldable solar panel based on origami concept. In medical applications, origami has inspired the design of new medical apparatus especially dedicated to minimally invasive surgery.
Shape memory alloys (SMAs) are widely employed as actuators due to their capacity of
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generate high forces and large displacements when subjected to thermomechanical fields that yield phase transformations. The association of SMA actuators with origami structures is
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promising being the focus of some research efforts (Koh and Cho, 2013; Pesenti et al., 2015; Kim et al., 2016). Lee et al. (2013) developed an origami-wheel actuated by SMA connected to a bias
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linear spring, being able to alter its radius in a robotic vehicle. The waterbomb pattern is employed for the origami design as employed on several applications since it allows significant shape changes (Hanna et al., 2014; Ma and You, 2014). Le et al. (2014) also used a waterbomb
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origami pattern in an aerial drone employed to landing in water. Salerno et al. (2014) presented an SMA origami device that takes advantage of geometric relations of the origami movements to
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increase the maneuver capacity. Origami system is a slender structure with complex geometry that makes stability analysis an
essential part of the design. Therefore, dynamical behavior of origami systems is an important issue to be investigated and only few works are available in literature. Basically, this analysis combines geometric and constitutive nonlinearities, providing a rich dynamics with operations close to the stability limits. In this regard, Fonseca et al. (2016) presented a dynamical analysis of
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an origami-wheel with shape memory actuators. A single degree of freedom system is employed as an archetypal model of the origami structure. Results show a rich response, including chaos. This work investigates the nonlinear dynamics of an origami-stent, a self-expandable cylindrical origami being originally presented by Kuribayashi et al. (2006). The stent is usually employed as a tubular medical device used to protect weakened arterial walls in the human body,
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but it can also be applied to increase stiffness of several systems including pipes and oil drilling well. Basically, origami-stent has a cylindrical shape and the folds allow the change of its radius
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(Figure 1).
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Figure 1 – Origami-stent deployment: three different configurations from folded, closed one, to
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deployed, opened one.
A shape memory alloy origami-stent system is analyzed considering torsion SMA wires
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(TSW) as actuators. A polynomial constitutive model describes thermomechanical behavior of SMAs. Origami geometric relations are employed to propose a one-degree of freedom (1-DOF)
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system that represents the system dynamics. Numerical simulations are carried out showing rich responses due to situations related to different thermomechanical operational excitations.
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After this introduction, this paper is organized as follows. Initially, a geometric analysis is
performed, establishing geometric relations and the kinematics of the origami system. Dynamical model is then proposed considering a single-degree of freedom system. Afterward, numerical simulations are discussed considering different thermal and thermomechanical loadings. Concluding remarks are presented in the sequence.
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2 - GEOMETRIC ANALYSIS The origami-stent is a self-expandable cylindrical origami constructed by following the folding pattern showed in Figure 2-a, where the continuous lines mean mountain folds (folds outwards the paper plane) and the dashed lines mean valley folds (folds inwards the paper plane). This pattern is composed by the repetition of rectangular components as showed in Figure 2-b,
side. The rectangular element has a length
and an angle
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which is known as waterbomb base, shifted half an element in comparison with those on their , and therefore,
⁄
and
means it is a square element. A single waterbomb base folded is presented in Figure 2-
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c.
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Figure 2 – Origami-stent folding pattern: a) waterbomb folding pattern where continuous line represents mountains and dashed line represents valleys; b) plane view of the single cell element;
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c) three-dimensional view of the single cell element.
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Geometric relations are now in focus and some hypotheses are established for this aim. The
first one is the symmetry, assuming that all elements behave in the same way and the element itself behaves symmetrically. Under this assumption, one-quarter of the cell element is representative of the general cell behavior. The second assumption is that fold occurs only in the creases and therefore, the element facets remain straight. Under these assumptions, it is possible to analyze the whole origami from a single cell element.
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The deployment of the origami goes through two stages (Kuribayashi, 2004). On the first stage, the element unfolds until it reaches a flattened configuration and the nodes A, C and A2 are
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co-linear. On the second stage, the crease AA2 moves outwards in the radial direction (Figure 3).
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Figure 3 – Origami-stent two-stage configurations: a) row of elements;
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b) single element.
The cell element geometry is characterized by three angles: ,
and
. These angles are
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coupled in such a way that each one can be described as a function of the others. Hence, it is possible to elect
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while
as the basic angle. It is noticeable that
characterizes the first stage
characterizes the second stage.
On the first stage,
can be expressed by considering the positions of the points A and B on
the (x,y,z) system with origin on C: A = (
) and B = (
).
Since the distance AB is equal to , the following equation can be written: (1)
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By solving Equation (1), the following equation is obtained: )
Based on that, there are two possible solutions for
(2) . Since this angle cannot be constant on
the first stage, the following equation is established to define ( In order to analyze the angle Since ApC =
(3)
)
(4)
, it is possible to write ApDp = and, since the angle ∠ADDp =
.
, the distance ApDp
. Therefore, the following relation is written for the first stage: ⁄
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(
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=
)
on the first stage, consider the triangle ApBpC (Figure 4).
and DpC =
On the other hand, AD =
during the first stage:
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(
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Figure 4 – Geometric characterization using angles of the origami-stent single element.
In order to obtain a function of
that depends only of , the angle
AC
substituted by the Equation (3) and it is found that On the second stage, Figure 3 shows that the angle
in the Equation (4) is .
remains constant, being equal to
.
This can be also pointed out by considering the second term on the left side of Equation (2). Besides, since both facets that contains the crease BC remains on the same plane,
.
An analysis of the origami-stent radii is now of concern. Let O be the central point of the origami-stent. Figure 5 shows two radii that define origami geometry: an internal radius,
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,
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, defined as OBp. The angle , defined as ∠ApOBp, is
defined as OC; and an external radius,
related to the number of cell elements circumferentially distributed, n, and by the hypotheses already assumed, it is equal to all elements. Therefore, this angle is such that
.
Hence: (5) is the angle ∠ApCBp, it is possible to see from Figure 5 that
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Reminding that
, furnishing the following relations:
(6) (7)
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and
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Figure 5 – Geometric characterization using internal and external radii of the origami-stent:
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a) perspective view; b) frontal view.
The origami has physical limitations that can be expressed as follows: the radius
AC
less than zero, which means that the structure does not penetrate itself;
cannot be
cannot be greater than
, which means that the mountain AC does not become a valley avoiding a fold in another
direction that would result in the structure collapse. Hence, and
⁄
implies that
the definition of limit angles,
and
implies that
. These restrictions can be written by
: (8)
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(9) In summary, the origami geometry can be described with the following set of equations: {
(10) )
(11)
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(
{
Based on the geometric relations of the origami-stent, Figure 6 illustrates the general behavior
and
45°). Figure 6-a shows the evolution of the angles
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of an origami with square elements (
as a function of the angle . Note that at
, both angles ( and ) have a transition
configuration between the different stages. Figure 6-b shows internal and external radii divided by the length L for different values of n (the number of elements circumferentially distributed). Besides the limit angles,
and
, the number n influences the radius maximum values. occurs at
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The maximum value of the external radius
degree of compactness of the structure. For n = 6,
. Figure 6-b also shows the
can double its size, while for n = 10,
can
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triplicate its size. Hence, the greater is the value of elements n, the more compact the origami is able to be. Nevertheless, the increase of the number of elements makes harder the origami
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manufacturing process.
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90
60
30
0 30
45
60
()
75
90
Re
n = 10
Ri
n=8
2
n=6
1
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Radius / L
Angle (°)
3
0
105
15
30
45
60
75
90 105 120
Figure 6 – Geometric relations defined by origami angles and radius as a function of degrees for
= 45°): a) angle relations ( and ); b) radius relations (
) for different
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values of n.
and
(in
2.1 - Kinematic Analysis
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Kinematics of the origami-stent system is now treated. Due to symmetry assumptions, it is possible to analyze one-quarter of the cell element. A global coordinate frame F(OXYZ) is placed
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on the center axis of the origami-stent with origin on O (see Figure 5). A local coordinate frame P(Cxyz) is placed due to a translation from O to C (see Figure 5). Hence, the coordinate frame P is translated from the coordinate frame F through Z direction of a distance equal to OC, that is,
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.
Basically, kinematic analysis corresponds to two triangles which movements are
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geometrically coupled to each other. It is assumed that each one of these triangles rotates presenting inertia. The triangle BBpC rotates of an angle and then of an angle
AC
angle
(see Figure 4). Besides that, translation motion associated with
each triangle has a lumped mass placed in its centroid: and
while the triangle ABC rotates of an
on the centroid of the triangle ABC
on the centroid of the triangle BBpC.
Therefore, five coordinate frames are considered: F and P; Q, R and S. The coordinate frames Q, R and S have origin on C, presenting movements related to each geometric angle, , respectively (Figure 7). The triangle ABC rotates about
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and ,
and, then, rotates about
.
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Q(Cx(1)y(1)z(1)) rotates about
in the x(1)-axis direction of P while R(Cx(2)y(2)z(2)) rotates about
in the z(2)-axis direction of Q. The triangle BBpC rotates . S(Cx(3)y(3)z(3)) rotates about
on the
y(3)-axis of P. Since the coordinate frame P does not rotates about the global coordinate frame F (it only translates), the rotation matrices are the following: [
]
[
]
(13)
[
]
(14)
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(12)
Figure 7 – Coordinate frames defined on the origami-stent single element.
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It is assumed that the lumped masses
and
are located in the geometric center of the
triangles ABC and BBpC, respectively. The vectors that indicate their positions are represented on the coordinate frames R and S, respectively, given by: ]
[
]
(15)
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[
(16)
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The triangle inertia tensors are the following:
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[
]
(18) ]
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[
(17)
After these definitions, it is defined triangle angular velocities with respect to the global
̇
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The lumped mass linear velocities
̇+ ̇
*
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frame:
(19)
[ ̇]
and
are given by the position time derivative,
reminding that all angles are described as a function of
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(20)
.
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[
] ̇ ̇ ̇
[
(
(21)
* ̇
(
* ̇
̇
]
[
]
[
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̇
]
̇ 3 - DYNAMICAL MODEL
(22)
̇
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The dynamical analysis of the origami-stent system assumes kinematic relations in order to write an equivalent 1-DOF system to represent the origami dynamics. Equations of motion are formulated by considering energetic approach, electing angle
as a generalized coordinate.
Using the transformation matrices to describe all terms in the system F, the kinetic energy
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given by:
(23)
)
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(
Figure 8 shows the kinetic energy E as a function of
is
and its time derivative, ̇ , for a square
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element, considering n = 6, unitary length L and unitary masses. Figure 9 shows the projection of this energy for different values of n, in the left as a function of ̇ for for a unitary velocity ( ̇
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function of
1°/s). It should be pointed out that the energy is a
90° due to changes of the angles
AC
nonsmooth function at
= 90° and in the right as
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and
(see Figure 6).
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Figure 8 – Kinetic energy of origami-stent square element with unitary length and unitary masses
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5
0 -1.0
0.0
0.5
1.0
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-0.5
E (10-3 J)
10
n=6 n=8 n = 10
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E (10-3 J)
10
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and six elements (n = 6).
n=6 n=8 n = 10
5
0
15 30 45 60 75 90 105
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Figure 9 – Projections of the kinetic energy of origami-stent square element with unitary length = 90°; b) ̇
1 °/s.
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and unitary masses for different number of elements, n; a)
The actuation of the origami-stent is provided by a Torsion SMA Wire (TSW) with length
and radius
(Koh et al., 2014). Two TSWs are antagonistically placed on each side of a cell
element on the crease AC (Figure 10) and these actuators promote the origami configuration changes (opening and closing).
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Figure 10 – Actuation system provided by Torsion Spring Wires (TSWs): a) the placement of the actuators at the planar single element; b) a folded single element with TSWs and lumped masses
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placed on the triangles’ centroids; b) Torsional Spring Wire (TSW).
The SMA thermomechanical behavior is described by a polynomial constitutive model (Falk, 1980; Savi & Pacheco, 2002) that establishes a stress-strain-temperature
relation.
This constitutive model considers that the austenitic phase is stable at high temperatures, and two variants of the martensitic phase, induced by positive and negative stress fields, stable at low
,
and
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where
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temperatures.
(24)
are material parameters and
as the temperature above which the austenitic
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martensitic phase is stable. By considering
is the temperature below which the
phase is stable, it is possible to write the following relationship:
.
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Aguiar et al. (2010) and Enemark et al. (2016) showed that helical spring force-displacementtemperature curve is similar to the stress-strain-temperature curve assuming homogeneous phase
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transformation through the spring element. Under this assumption, it is possible to use the same argues to define an torsional actuator torque-displacement-temperature relation that is formally similar to the stress-strain-temperature: * (
( )
) (
15
) ]
(25)
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where
is the angle where the TSW is free of stress. Note that this expression represents
constitutive relation of both actuators, being expressed by
and
On the origami-stent system, the generalized forces
are related to the TSW generalized
, which due to symmetric considerations is given by:
An external generalized force the virtual work
.
is applied at the point C, z-direction. Hence, the expression of
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force
.
, neglecting gravitational effect, is given by:
(26)
Note that
⁄
,
, and therefore,
⁄
.
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Two extra generalized forces are considered in order to have a complete description of the origami-stent system. The first one is related to a linear viscous dissipation of coefficient ̇
representing all dissipation processes, including the hysteresis phenomenon: second one is related to origami geometric restrictions that limit
. The
to vary in the range [θmin, θmax]
(27)
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{
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(Equations (8) and (9)), expressed as a moment of the form:
where klim limits the system response and represents an equivalent elastic spring stiffness.
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Equation of motion is then obtained from Lagrange’s equation. ̇
( )
(28)
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( ̇)
The kinetic energy (23) is a nonlinear expression in
multiplied by the velocity and can be
̇ . Details about this function are presented in Appendix. Substituting this
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written as
expression in the left hand side of Lagrange’s equation (28), it gives a term
̈ and a term
̇ (see Appendix for more details). Therefore, the equation of motion is written as follows: ̈
̇
̇
(29)
In order to write dimensionless equations, it is defined a dimensionless time where
is a reference frequency, defined from the linear part of the restitution force:
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,
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√
(30)
Hence, after dividing the Equation (29) by
, the following set of equations of
motion is written: { ⁄
where
, and therefore
̇
̈
and
dimensionless terms are employed:
)
(31)
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(
. Besides, the following
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(32)
(34) (35) (36)
Figure 11 presents dimensionless functions
and at
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masses. The discontinuity of the function
(33)
for a square element and equal
90° is due to the nonsmoothness of the
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kinetic energy. These functions indicate the origami-stent nonlinearities.
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1.0
0.15
0.10
f2
0.0
0.05
n=6 n=8 n = 10 -1.0 15 30 45 60 75 90 105 -0.5
Figure 11 – Dimensionless functions
0.00
and
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f1
0.5
n=6 n=8 n = 10
15 30 45 60 75 90 105
that define geometrical nonlinearities of = 45°).
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the dynamical model (
4 - NUMERICAL SIMULATIONS
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Dynamical behavior of the origami-stent system actuated by two antagonistic TSWs is now analyzed. Numerical simulations are carried out by employing the fourth-order Runge-Kutta
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method. Different thermal and thermomechanical loads are contemplated to study the system behavior under distinct operational conditions.
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Table 1 presents the origami-stent parameters employed in all simulations considering a square element. Besides, a dissipation coefficient
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the reference frequency is
= 0.3 is employed. Under these conditions,
rad/s. Temperature
represents a reference
temperature where the origami-stent actuator does not apply any moment at a chosen angle of
AC
(closed configuration). Thereby, one actuator has
. The other one has
since this assures that the origami can reach the opened configuration of maximum radius (
) by changing the TSW temperature. This set of parameters can represent a human body
stent. Note that the length =13.2 mm for
of the cell element is 6.6 mm. Hence, for n = 6, the external radius
= 90° (equation (6)), which is close to the usual radius of aortal stent grafts.
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Figure 12 presents the torque of the TSWs as a function of 310K (
). TSW-1 is located at
with a temperature
at the temperature
with a temperature
and at T =
while TSW-2 is at
.
(kg)
(kg) 510-5
510-5
(MPa/K)
(N m)
8.010-5
(MPa)
(MPa)
(K)
(K)
(K)
1.41010
2.21012
287.15
309.1
293.15
2.210-3
1.010-2
0.5
M
0.0 TSW-1 TSW-2 0
45
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-1.0 -90 -45
90
135 180
M (10-6 N m)
1.0
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M (10-6 N m)
(m)
6.610-3
1.0
-0.5
(m)
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1.0106
(m)
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Table 1 – Origami-stent parameters.
0.5 0.0
-0.5
TSW-1 TSW-2
-1.0 -90 -45
0
45
90
135 180
Figure 12 – Torque of the TSWs at different temperatures: a) reference temperature
;
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b) T = 310 K.
AC
Concerning external excitation, a harmonic external force is considered of amplitude
and
frequency .
(37)
In addition, temperature has fluctuations represented by: (38)
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where
is the temperature around which the temperature
oscillates.
4.1 - Thermal Loadings The actuator temperature influence on the system dynamical response is investigated considering different thermal loadings. An overview of the origami-stent equilibrium
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configurations can be provided by the analysis of the moments represented by (Equation (34)). Figure 13 presents these moments for different pairs of temperatures and n = 6. In the reference configuration (Figure 13-a),
=
three angles: 30°, 60° and 90°. If the temperature
=
, the sum of the moments vanishes at = 310 K) while
=
, only 30° is an equilibrium point (Figure 13-b). On the other hand, if the temperature
is
(
= 310 K) while
=
(
, only 90° is an equilibrium point (Figure 13-c). If the
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greater than
is greater than
temperatures of both TSWs are greater than
(
=
= 310 K), there are again three points
where the sum of the moments is zero (Figure 13-d). Therefore, by controlling the temperature one can establish the configuration (opened or closed) of the origami-stent, defining the
AC
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equilibrium point structure.
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0.08
0.04
0.04
H1
H1
0.08
0.00
-0.04 -0.08
-0.04 30
45
60
75
90
-0.08
105
0.08
45
60
75
90
105
60
75
90
105
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0.08
0.04
H1
H1
0.04 0.00
0.00
-0.04
30
45
60
M
-0.04 -0.08
30
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0.00
75
90
105
-0.08
30
45
ED
Figure 13 – Actuator temperature influence on the generalized forces: a) TSW-1 and TSW-2 at reference temperature with three vanishing points; b) TSW-1 at high temperature with one
PT
vanishing point at 30°; c) TSW-2 at high temperature with one vanishing point at 90°; d) TSW-1
CE
and TSW-2 at high temperature with three vanishing points.
Free vibration analysis is now of concern identifying the temperature influence. Figures 14
AC
and 15 present the dynamical behavior of the system for different initial conditions and temperatures. The temperature of one actuator is modified while the other remains constant at the reference temperature. Figure 14 presents the phase space at different temperatures for the TSW1 while TSW-2 temperature is at
=
. The sum of the moments points up to three equilibrium points
, and the phase space shows that two of them are stable while the other one is unstable.
By increasing the temperature
(
= 306 K), 30° remains a stable equilibrium point while the
21
ACCEPTED MANUSCRIPT
other stable equilibrium point is, now, smaller than 90°. When
>
(
= 310 K), only 30° is
an equilibrium point, meaning that the structure admits only the closed configuration. 5.0
0.08
H1
0.04 0.00
-0.04 5.0 30
45
60
75
90
v (degree) v (degree)
30
30
45
45
60
75
(degree) 60
75
(degree)
90
105
M
-2.5 -5.0
90
60
105
75
(degree)
60
75
(degree)
AN US
0.0 -2.5
45
45
5.0 2.5
2.5 0.0
-5.0
30
-4 5.0 30
105
ED
v (degree) v (degree)
5.0 2.5
0 -2.5 -2 -5.0
T1 = T 0 T1 = 306 K T1 = 310 K
-0.08
2 0.0
CR IP T
v (degree) v (degree)
4 2.5
90
90
105
105
2.5 0.0
0.0 -2.5
-2.5 -5.0 -5.0
30
30
45
60
75
(degree)
45
60
75
(degree)
90
90
105
105
Figure 14 – Origami free vibration response at different temperatures with 6 elements (n = 6). and different values of
PT
a) Sum of the moments at
CE
space at
. b) Phase space at
. d) Phase space at
. c) Phase
.
Figure 15 presents the phase space at different temperatures for the TSW-2 while the . By increasing the temperature
(
= 306 K), 90° remains a
AC
temperature of the TSW-1 is
stable equilibrium point while the other stable equilibrium point is now, greater than 30°. When >
(
= 310 K), only 90° is an equilibrium point, and the structure only admits an opened
configuration. It is important to highlight that the origami-stent is initially in a reference configuration where the structure is closed ( = 30°). The opened configuration can be reached either by temperature variation or by imposing proper initial conditions, as one can see in Figure
22
ACCEPTED MANUSCRIPT
15-a (or Figure 14-a). This reveals an initial condition dependence that is now investigated using different thermal loads.
5.0
0.08
H1
0.04 0.00
-0.04 45
60
75
90
v (degree) v (degree)
4 2.5
45
30
45
60
75
(degree)
90
ED
30
M
0 -2.5
-4
-4 5.030
105
45
45
60
75
90
(degree)
60
75
90
(degree)
105
105
4 2.5
2 0.0
-2 -5.0
30
AN US
30
0 -2.5 -2 -5.0
T2 = T 0 T2 = 306 K T2 = 310 K
60
75
(degree)
v (degree) v (degree)
-0.08 5.0
2 0.0
CR IP T
v (degree) v (degree)
4 2.5
105
90
105
20.0
0 -2.5
-2 -5.0 -4
30
30
45
45
60
75
90
(degree) 60
75
(degree)
90
105
105
PT
Figure 15 – Origami free vibration response at different temperatures with 6 elements (n = 6). a) Sum of the moments at
CE
space at
and different values of
. d) Phase space at
From the reference condition, when temperature
AC
. b) Phase space at .
increases up to 310 K, the origami-stent
opens (Figure 16). System response presents an oscillation before reaching temperature
decreases to
increasing temperature By decreasing
to
. c) Phase
= 90°. If the
again, the origami-stent remains in the opened configuration. By
, 90° is not an equilibrium point anymore and the origami-stent closes.
, the structure remains closed.
23
ACCEPTED MANUSCRIPT
315
105
T2
310
90
305
75
(°)
300
45
295 290
60
30 0
2000
4000
6000
8000
0
CR IP T
(K)
T1
2000
4000
6000
8000
Figure 16 – Opening-closing process induced by temperature variations (n = 6): a) thermal
AN US
load; b) time history of the deployment angle θ.
A different thermal loading is now investigated. By increasing subsequently increasing
up to 310 K and
up to 310 K (Figure 17), the structure remains opened but not
completely, since the equilibrium point is around 82°. By decreasing the temperature there is no more changes.
T1
305 300
CE
295
0
AC
290
90
PT
(K)
310
105
T2
(°)
315
ED
M
origami-stent closes and by decreasing
, the
2000
75 60 45 30
4000
6000
0
8000
2000
4000
6000
8000
Figure 17 – Opening-closing process induced by temperature variations (n = 6): a) thermal load; b) time history of the deployment angle θ.
24
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is first increased (Figure 18), there is no structure change. By increasing
subsequently, there is a small increase of the angle the stent-origami opens and, subsequently, if
90
305
75
(°)
(K)
105
T2
310
300
60 45
295 290
30 0
2000
decreases,
decreases, it remains opened.
315
T1
and it stabilizes around 38°. If
4000
6000
0
2000
4000
AN US
8000
CR IP T
In contrast, if
6000
8000
Figure 18 – Opening process induced by temperature variations (n = 6): a) thermal load; b) time
M
history of the deployment angle θ.
Equilibrium configuration small increase can be obtained increasing both temperatures
ED
simultaneously as presented in Figure 19. Note that the angle 315
T1
90
300
(°)
PT
305
AC 0
75 60 45
295 290
is changed to 38°.
105
T2
CE
(K)
310
and
30
1000
2000
3000
0
1000
2000
3000
Figure 19 – Opening process induced by temperature variations (n = 6): a) thermal load; b) time history of the deployment angle θ.
25
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Until now, all simulations consider the same number of elements, n = 6. The change of the number of elements can influence the dynamical response, which is now investigated. The first point that should be observed is that when n changes, the limit angles change; both tend to reduce for greater values of n. For n = 6,
= 30° and
and
= 105°. For n > 6,
< 30° and the origami is not completely closed at the reference condition and
< 105°.
Figure 20 presents the origami response when temperature
CR IP T
Besides that, Figure 9 shows that the amount of kinetic energy is larger for greatest values of n. increases from the reference
condition causing the opening of the origami. Different values of n are treated. Although the actuation response occurs at the same time, the stabilization takes more time to occur for origami with larger number of elements due to the greater amount of kinetic energy.
T1
AN US
315
105
T2
90
(°)
305 300
290
75 60 45
295
0
1000
2000
3000
M
(K)
310
4000
n=6
30
5000
0
1000
2000
90
(°)
60 45
AC
1000
75 60 45
n=8
30
0
5000
105
PT
75
CE
(°)
90
4000
ED
105
3000
2000
3000
4000
n = 10
30
5000
0
1000
2000
3000
4000
5000
Figure 20 – Comparison of the opening process considering different values of the number of elements, n. a) thermal load; time history of the deployment angle θ for: b) n = 6; c) n = 8; d) n = 10.
26
ACCEPTED MANUSCRIPT
Since the origami system has strong temperature dependent behavior, it is important to evaluate the effect of thermal oscillations on system response. Figures 21 to 23 present the influence of thermal oscillations on temperatures
. Afterward, both temperatures oscillate at the
opened configuration. A variation of temperature the temperature
considering n = 6. Figure 21 presents
does not alter the response, but a variation of
CR IP T
the opening process changing temperature
and
causes small oscillations. The TSW-2 is at a high temperature and 90° is a
free-stress point (see Figure 12) that does not change with the thermal variation. Based on that, the sum of system moments is not altered and, therefore, its equilibrium point does not change. However, the TSW-1 is at the reference temperature and thermal fluctuation alters the free-stress
oscillates while the temperature
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points, except the one at 30°. Hence, the sum of the moments is altered and its equilibrium point oscillates.
315 T1
105
T2
M
305 300 295 290
0
1000
2000
3000
4000
(°)
90
ED
(K)
310
75 60 45 30
5000
0
1000
2000
3000
4000
Figure 21 – Opening process with thermal oscillation (n = 6): a) thermal load (
= 2.0 K and
= 0.1); b) time history of the deployment angle θ.
AC
CE
5000
PT
Figure 22 shows a similar opening condition but with a situation where both
increased. Under this assumption, a variation of temperature
and
are
also induces oscillations and the
origami response is a combination of both influences. Since both actuators have
, a
variation of any actuator temperature is responsible to alter the sum of the moments of the system and, as a consequence, there is an oscillatory response. Besides, frequency thermal frequency
27
ACCEPTED MANUSCRIPT
phase is analyzed. It is noticeable that when actuator thermal variation is out of phase, the oscillatory response has higher amplitude.
315
90
305
75
310
300
60
295
45
290
30 0
1500
3000
4500
315 T1
6000
0
1500
3000
4500
6000
3000
4500
6000
105
T2
310
90
305
75 60
300
M
(K)
CR IP T
105
T2
AN US
(K)
T1
295 0
1500
3000
4500
ED
290
6000
45 30 0
1500
Figure 22 – Opening process with thermal oscillation (n = 6): a) and c) thermal load (
= 2.0
PT
= 0.1); b) and d) time history of the deployment angle θ.
CE
K and
Figure 23 presents a situation where both actuators have
AC
configuration. Under this condition, only the variation of contrast of the case of Figure 21, small fluctuation of
, characterizing a closed affects the dynamic response. In
does not alter actuator dynamical
response since 30° is an actuator free-stress point (see Figure 12) that does not change with a thermal variation. On the other hand, TSW-2 has 90° as a free-stress point that does not change, while the other points change with the temperature and, hence, thermal fluctuation changes the sum of the moments, affecting the system equilibrium point.
28
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315
105
T2
310
90
305
75
(°)
300
45
295 290
60
30 0
500
1000
1500
2000
2500
0
500
CR IP T
(K)
T1
1000
1500
2000
Figure 23 – Closed configuration with thermal oscillation (n = 6). a) Thermal load (
= 2.0 K
= 0.1); b) time history of the deployment angle θ.
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and
2500
The influence of the number of elements in a case with thermal fluctuation is presented in Figure 24. Note that the increase of the number of elements, n, tends to cause a greater change on
AC
CE
PT
ED
M
system response.
29
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315
105
T2
310
90
305
75
(°)
300
45
295 290
60
n=6
30 0
1500
3000
4500
6000
0
1500
3000
4500
6000
105
90
90
75
75
(°)
105
60 45
AN US
(°)
CR IP T
(K)
T1
60 45
n=8
30 0
1500
3000
4500
0
1500
3000
4500
6000
M
6000
n = 10
30
Figure 24 - Comparison of the opening process with thermal oscillation considering different for:
ED
values of the number of elements, n. a) Thermal load; time history of the deployment angle
PT
b) n = 6: c) n = 8; d) n = 10.
CE
4.2 - Thermomechanical Loadings Thermomechanical loadings are now addressed in order to simulate some operational =
AC
conditions of the origami-stent. Basically, opened configuration is of concern considering and
= 310 K. Under this condition, only 90° is an equilibrium point and an external force is
applied. In order to establish a physical interpretation, the origami-stent system is assumed to represent human body stent. Under this assumption, the magnitude of the external force is compatible to the human body blood pressure. Hence, the dimensionless amplitude external force
is in a range from 0.01 to 0.85, which is related to force of magnitude
30
of the N.
ACCEPTED MANUSCRIPT
Initially, a global analysis is performed considering bifurcation diagrams that present a stroboscopic view of system dynamics under the slow quasi-static variation of parameter
for
0.30 and different number of elements: n = 6; n = 8; n = 10 (Figure 25). Figure 25-a considers n = 6. Note that, for small force amplitudes, the system presents period-1 response until = 0.143. After that, a cloud of points suggests a chaotic response. By increasing the value of ,
When
CR IP T
the diagram shows other regions where the system response has a periodicity greater than 1. = 0.156, bifurcations start from chaotic-like response to period-1 response. A period
doubling can be noticed at
= 0.260, nevertheless a period-1 response is predominant until
=
0.628, where the system response becomes chaotic. Inside this chaotic region, there are some periodic windows, especially between
= 0.740 and
= 0.781.
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By considering different number of elements, n = 8 (Figure 25-b) and n = 10 (Figure 25-c), it is noticeable that the increase of the number of elements tends to stretch the diagram and spread the kind of behavior over the values of . Note, for instance, that period-1 response for small amplitudes of the force finishes at
= 0.143 when n = 6 but at
= 0.157 when n = 8 and at
=
0.308 when n = 10. After the region of period-1 response, different bifurcation evolutions occur
M
for each cell number. When n = 6, the cloud of points occurred immediately after the period-1 response. When n = 8, a period doubling takes place until the system response is quasi-periodic
ED
and then, the cloud of points takes place. When n = 10, a period doubling occurs then the cloud of points takes place. Subsequently, a periodic response window is noticed until there is a response
PT
sudden change suggesting a chaotic response. Afterward, a bifurcation from chaos to period-1
AC
CE
occurs.
31
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(degree)
105 75 45
0.0
0.2
0.4
0.6
75
75
15 n = 8
45 15
0.2
0.4
0.6
0.8
M
0.0
0.8
AN US
(degree)
105
(degree)
105
45
CR IP T
15 n = 6
n = 10
0.0
Figure 25 - Bifurcation diagrams varying excitation amplitude
0.4
0.6
0.8
for different number of elements
0.30): a) n = 6; b) n= 8; c) n = 10.
ED
and a fixed frequency (
0.2
PT
Phase space and its respective Poincaré section are presented in Figure 26 for different amplitudes of the external force taken from the bifurcation diagram of n = 6. In all these cases,
CE
the external force is great enough to push the origami to its physical limits, which is indicated by the straight lines at 30° and 105°. For
= 0.145, a point inside the cloud of points region, a
AC
chaotic response occurs. By increasing values of , period-1 and period-2 responses are achieved for
= 0.20 and
= 0.36, respectively. A chaotic response is again obtained for
32
= 0.70.
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5.0
5.0 16
16
2.5
(degree) vv(°)
0.0 0
-8
-5.0 30 -16 15 30 5.0
-5.0
45 45
60 75 90 105 60(degree) 75 90 105 120
(°)
45 30
45
v (°) v (degree)
30 2.5
10 0.0 0 -2.5
15 0.0
60
75
(degree) 60
75
(°)
AN US
v (°) v (degree)
30
-16 15 5.0
20 2.5
90
90
105
105
120
0 -2.5
-15
30 30
45 45
60
75
90
(degree) 60
75
(°)
90
105
105
-5.0 30 -30
120
15
M
-20 15
0
-2.5
-2.5 -8
-10 -5.0
8
0.0
CR IP T
vv (degree) (°)
2.5 8
Figure 26 – Origami-stent response for
30
45
60
75
90
(degree) 60
75
(°)
90
105 105
120
= 0.3 and n = 6. Phase space (black lines) and
= 0.145. b)
= 0.200; c)
= 0.360. d)
= 0.700.
ED
Poincaré section (red points). a)
45
PT
The same results of Figure 26, that considers n = 6, is now presented for n = 8, Figure 27, and n = 10, Figure 28. When
= 0.145 (Figures 27-a and 28-a), the system oscillates around 90° with
CE
a period-1 response, without reaching the physical limits of the structure. When
= 0.20, there is
a quasi-periodic response for n = 8 and, even if the system still oscillates around 90°, the upper
AC
physical limit is reached. In contrast, for n = 10, this increase of response substantially. On the other hand, when 8, but chaotic for n = 10. When
does not affect the system
= 0.36, the system response is periodic for n =
= 0.70, a period-1 response is noticed.
33
5.0
5.0
4 2.5
4 2.5
2 0.0 0 -2.5 -2 -5.0
30
45
-4 60 5.0
60
75
90
(degree) 75
(°)
v (°) v (degree)
0 -2.5
30 30
45 45
60
75
90
(degree) 60
75
(°)
90
105
105
Figure 27 – Origami-stent response for
120
75
90
(°)
90
105 105
0 -2.5
-16 15
30
30
45
45
60
75
90
(degree)
60
75
(°)
90
105
105 120
= 0.3 and n = 8. Phase space (black lines) and
= 0.145. b)
AC
CE
PT
ED
Poincaré section (red points). a)
8 0.0
-8 -5.0
M
-16 15
60
(degree)
75
16 2.5
8 0.0
45
AN US
v (°) v (degree)
30
-4 60 5.0
105
16 2.5
-8 -5.0
0 -2.5 -2 -5.0
105
90
2 0.0
CR IP T
v (degree) v (°)
v (degree) v (°)
ACCEPTED MANUSCRIPT
34
= 0.200; c)
= 0.360. d)
= 0.700.
5.0
5.0
4 2.5
4 2.5
2 0.0 0 -2.5 -2 -5.0
30
45
-4 60 5.0
75
60
75
90
(degree) 90
(°)
105
30
45
-4 60 5.0
105
75
v (°) v (degree)
20 2.5
10 0.0 0 -2.5
10 0.0
30
45 45
75
90
60
75
(°)
90
105 105
120
M
30
60
(degree)
Figure 28 – Origami-stent response for
90
105
90
(°)
105
-20 15
30
45
30
45
60
75
90
(degree) 60
75
(°)
90
105 105
120
= 0.3 and n = 10 elements. Phase space (black lines)
= 0.145. b)
= 0.200; c)
= 0.360. d)
= 0.700.
ED
and Poincaré section (red points). a)
75
0 -2.5
-10 -5.0
-20 15
60
(degree)
AN US
v v(°)(degree)
0 -2.5 -2 -5.0
20 2.5
-10 -5.0
2 0.0
CR IP T
v v(°)(degree)
v v(°)(degree)
ACCEPTED MANUSCRIPT
PT
Resonant condition is usually critical to system response. Concerning chaotic behavior, they can be induced more effectively close to this condition. In this regard, Figure 29 presents the
CE
maximum amplitude of displacement under the variation of frequency . For n = 6 (Figure 29-a), it presents a typical resonant curve with maximum amplitude at
0.085, related to constitutive nonlinearities. For n = 8 and n = 10, the peaks of
AC
small peak at
0.159. It is also noticeable a
maximum angle is shifted to the left and the flat region related to the maximum values is due to the maximum displacement constraints imposed by the origami physical limits. Besides, the smaller peak is more pronounceable. Furthermore, Figure 29-d presents a zoom in Figure 29-c where dynamical jumps for different frequencies are observed.
35
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105
105
n=6
100
max
95 90 0.01
95 90
0.06
0.11
0.16
0.21
0.01
CR IP T
max
100
n=8
0.06
100.0
105
n = 10
max
95 90 0.01
0.06
0.11
97.5
AN US
max
100
0.11
0.16
0.21
M
Figure 29 – Origami-stent frequency diagram for
95.0 0.085
0.090
0.095
0.16
0.100
0.21
0.105
= 0.01 and different number of elements: a) n
= 6; b) n = 8; c) n = 10; d) enlargement of the resonance region for n = 10 elements showing
PT
ED
dynamical jumps.
Figure 30 shows the bifurcation diagram for n = 6 varying
under resonant condition (
CE
0.16). Note that system response is predominantly chaotic indicating the influence of the resonant condition on system dynamics. Phase space and Poincaré section is presented in Figure 31 for 0.16 and
= 0.20. A chaotic-like response is observed presenting a strange attractor in
AC
Poincaré section.
36
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75 45 15 n = 6 0.0
0.2
0.4
0.6
CR IP T
(degree)
105
0.8
20
5
v
0
M
v
0.16 and for n = 6.
10
10
30
45
ED
-10 -20
for
AN US
Figure 30 – Bifurcation diagram varying the excitation amplitude
60
75
-5 -10 30
105 ,
45
60
75
90
0.16 and n = 6. a) Phase space. b) Poincaré
section.
CE
PT
Figure 31 – Chaotic response for
90
0
AC
5 - CONCLUSIONS This paper deals with the nonlinear dynamical analysis of an origami-stent actuated by
shape memory alloys. A polynomial constitutive model is employed to describe the thermomechanical behavior of SMA actuators. Geometrical relations allow one to build a single degree of freedom system to analyze the system dynamics. This system has both constitutive and geometrical nonlinearities, presenting nonsmooth characteristics. Numerical simulations are carried out showing different thermomechanical loading situations representing operational 37
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conditions. Concerning thermal loadings, it is possible to identify that different configurations can be reached by changing the temperature of each actuator. Results are affected by the order of actuation. Moreover, thermal fluctuations can affect the system response depending on conditions. It depends on either the origami configuration or the actuator temperature. The number of cell elements interferes on the response stabilization time as well. An origami with
CR IP T
more cell elements takes more time to stabilize, being more affected by thermal fluctuations. Thermomechanical loadings can produce complex responses including chaos. The number of cell elements is critical in order to define the system response. In general, the increase of the number of cell elements reduces chaotic regions. Furthermore, greater values of external force are needed to push the origami to reach its physical limits. Dynamical jumps are also observed on origami-
6 - ACKNOWLEDGEMENTS
AN US
stent dynamics.
The authors would like to acknowledge the support of the Brazilian Research Agencies
M
CNPq, CAPES and FAPERJ. The Air Force Office of Scientific Research (AFOSR) is also
PT
7 - REFERENCES
ED
acknowledged.
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CE
investigations of shape memory alloy helical springs. Smart Materials and Structures, 19(2), 025008.
AC
[2] Enemark, S., Santos, I. F., & Savi, M. A. (2016). Modelling, characterisation and uncertainties of stabilised pseudoelastic shape memory alloy helical springs. Journal of Intelligent Material Systems and Structures, 1045389X16635845.
[3] Falk, F. (1980). Model free energy, mechanics, and thermodynamics of shape memory alloys. Acta Metallurgica, 28(12), 1773-1780.
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[4] Fei, L. J. & Sujan, D. (2012). Origami Theory and its Applications: A Literature Review. In 7th Curtin University Conference (CUTSE) Engineering Goes Green, Tang, F. E. et al (ed), pp. 336-340. Sarawak, Malaysia: Curtin University School of Engineering. [5] Fonseca, L. M., Rodrigues, G. V., Savi, M. A. & Paiva, A. (2016). Nonlinear Dynamics of an Origami Structure Coupled to Smart Materials. 6th International Conference on Science
and
Complexity
(NSC),
São
Paulo,
Brazil,
CR IP T
Nonlinear
doi:10.20906/CPS/NSC2016-0024.
[6] Hanna, B. H., Lund, J. M., Lang, R. J., Magleby, S. P. & Howell, L. L. (2014). Waterbomb base: a symmetric single-vertex bistable origami mechanism. Smart Materials and Structures, 23(9), 094009.
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[7] Kim, S. R., Lee, D. Y., Koh, J. S., & Cho, K. J. (2016). Fast, compact, and lightweight shape-shifting system composed of distributed self-folding origami modules. In Robotics and Automation (ICRA), 2016 IEEE International Conference on (pp. 4969-4974). IEEE. [8] Koh, J. S., & Cho, K. J. (2013). Omega-shaped inchworm-inspired crawling robot with large-index-and-pitch (LIP) SMA spring actuators. IEEE/ASME Transactions On
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Mechatronics, 18(2), 419-429.
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memory systems”, International Journal of Bifurcation and Chaos, v.12, n.3, pp.645-657.
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APPENDIX Origami-stent equations of motion are related to complex functions that express geometric relations. Kinetic energy definition (Equation (23)) is described on the global coordinate frame F, using transformation matrix. Therefore, consider that kinetic energy, Equation (22), can be
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written as follows: ̇
Applying this equation in the left hand side of the Lagrange’s equation (Equation (28)):
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̇
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Combining the terms with ̈ and ̇ , two others functions, (
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. The equation are functions of :
is presented in
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