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The Pennsylvania State University
The Graduate School
MODELING AND VALIDATION OF A COMPLIANT BISTABLE
MECHANISM ACUTATED BY MAGNETO ACTIVE ELASTOMERS
A Thesis in
Mechanical Engineering
by
Adrienne Crivaro
2014 Adrienne Crivaro
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Master of Science
August 2014
ii
The thesis of Adrienne Crivaro was reviewed and approved* by the following:
Mary I. Frecker
Professor of Mechanical Engineering
Thesis Co-Advisor
Timothy W. Simpson
Professor of Mechanical and Industrial Engineering
Thesis Co-Advisor
Paris Von Lockette
Associate Professor of Mechanical Engineering
Thesis Reader
Daniel Haworth
Professor of Mechanical Engineering
Professor-in-Charge of MNE Graduate Programs
*Signatures are on file in the Graduate School
iii
ABSTRACT
In the emerging field of origami engineering, it is important to investigate ways to
achieve large deformations to enable significant shape transformations. One way to achieve this
is through the use of bistable mechanisms. The goal in this research is to investigate the
feasibility and design of a compliant bistable mechanism that is actuated by magneto active
elastomer (MAE) material. The MAE material has magnetic particles embedded in the material
that are aligned during the curing process. When exposed to an external field, the material
deforms to align the embedded particles with the field.
First, the actuation of the MAE material through finite element analysis (FEA) models was
investigated. This helps predict the magnetic field required to snap the device from its first stable
position to its second for various geometries and field strengths. The FEA model also predicts
the displacement of the center of the mechanism as it moves from one position to the other to
determine if the device is in fact bistable. These results are validated using experimental models
and demonstrate the functionality of active materials to be used as actuators for such devices and
applications of origami engineering.
Next, parametric studies using the FEA model are performed to visualize the tradeoffs
between various design parameters. These results help show the relationship between the
substrate properties and the bistability of the device. With this information, it is possible to select
design parameters based on the desired arch displacement or allowable field strength for a
specific task.
iv
TABLE OF CONTENTS
LIST OF FIGURES ............................................................................................................... vi
LIST OF TABLES ................................................................................................................. viii
ACKNOWLEDGEMENTS ................................................................................................... ix
NOMENCLATURE .............................................................................................................. x
1 Chapter 1 Background and Motivation 1
1.1 Introduction 1 1.2 Literature Review 2
1.2.1 .. Origami Engineering 3 1.2.2 .. Bistablity 5 1.2.3 .. Bistable Compliant Mechanisms 6 1.2.4 .. Magneto Active Elastomer (MAE) Material 8
1.3 Research Objectives 10 1.4 Thesis Outline 10
2 Chapter 2 Finite Element Model of a Bistable Arch 12
2.1 Design Concept 12 2.2 FEA Model Set-up 12
2.2.1 .. Solid Mechanics Module 14 2.2.2 .. Magnetic Material Module 15 2.2.3 .. Study Steps 16 2.2.4 .. Determining Bistablity 19
2.3 Discussion 20
3 Chapter 3 Experimental Validation of the FEA Model 22
3.1 Experimental Validation 22 3.1.1 .. Experimental Set-up 22 3.1.2 .. Experimental Results 26
3.2 Discussion 28
4 Chapter 4 Parametric Variation of Bistable Arch 30
4.1 Parametric Sweep Set-up 30 4.2 Parametric Sweep Results 31 4.3 Bistable Region 34 4.4 Parametric Sweep Discussion 35
5 Chapter 5 Conclusions and Recommendations 37
v
5.1 Summary and Conclusions 37 5.2 Extension to Origami Engineering: A MAE-actuated Waterbomb 38 5.3 Future Work 39 5.4 Acknowledgments 42
Appendix A: Complete results for displacement and strain energy density 43
Appendix B: Extra images for experimental model 45
Appendix C: Complete results for the parametric Sweep 46
Bibliography .... 55
vi
1 LIST OF FIGURES
Figure 1-1: Example of the waterbomb .................................................................................. 1
Figure 1-2: A multi-field responsive origami structure actively folds from an initially flat
sheet to complex three-dimensional shapes in response to different applied fields. It
is also capable of actively unfolding from any shape back to the flat sheet. ................... 2
Figure 1-3: Deformable wheel robot [7] ............................................................................... 3
Figure 1-4: Ball on a hill analogy adapted from [22]. Positions A & C are stable; position
B is an unstable equilibrium position ............................................................................. 6
Figure 1-5: Residual-Compressive-stress buckled beam in its two stable positions.
Adapted from [23]. ......................................................................................................... 7
Figure 1-6: Pseudo-rigid-body model of a bistable arch (a) and the equivalent fully-
compliant system (b) from [31] ...................................................................................... 8
Figure 1-7: MAE self-locomotive device from[37] ................................................................ 9
Figure 1-8: A. Accordion constraints and remanent magnetization orientation adapted
from [37]. B. Arch constraints and remanent magnetization orientation for the work
being discussed. ............................................................................................................. 9
Figure 2-1: Initial shape, dimensions, and boundary conditions for COMSOL model ........... 14
Figure 2-2: Relationship between Ahi & magnetic field ......................................................... 15
Figure 2-3: Maxwell surface stress on the boundaries of the MAE patches ........................... 16
Figure 2-4: The initial displacement increases linearly with time .......................................... 17
Figure 2-5: Ahi increases linearly with time ............................................................................ 18
Figure 2-6: Ahi scalar as a function of time during the third study step .................................. 18
Figure 2-7: Steps used in the COMSOL model, 1 is bistable and 2 is nonbistable. A is the
initial shape, B is the initial stable arch shape, C is the initial application of the field,
D is the displacement as the field is applied, E is the position after the snap, and F
is the second stable position after the field is removed. .................................................. 19
Figure 2-8: Strain energy density of a bistable (top plot) and non-bistable (bottom plot)
design with corresponding steps shown in Figure 2-7 .................................................... 20
Figure 2-9: FEA prediction of center point displacement as the magnetic field increased ..... 21
vii
Figure 3-1: Test setup for experimental validation ................................................................. 23
Figure 3-2: Magnetic field as the voltage increases on the power supply .............................. 24
Figure 3-3: Experimental test setup with guidelines on the wood stand ................................. 25
Figure 3-4: Three samples for experimental validation .......................................................... 25
Figure 3-5: Image from the camera indicating the voltage and arch height ............................ 26
Figure 3-6: Plot of the center point height for the three test fixtures. The FEA model
results are also shown. .................................................................................................... 27
Figure 3-7: Stages of snap-through as the device is exposed to an increasing magnetic
field ................................................................................................................................ 28
Figure 4-1: A glyph plot in ATSV of the relationship between the thickness and initial
displacement of the PDMS substrate and the magnetic field required to cause snap-
through of the device design........................................................................................... 32
Figure 4-2: A glyph plot in ATSV showing the normalized MAE length variation and its
effect on the magnetic field required for snap-though .................................................... 33
Figure 4-3: A glyph plot in ATSV showing the normalized MAE separation variation and
its effect on the magnetic field required for snap-though ............................................... 34
Figure 4-4: Bistable region for the PDMS substrate design ................................................... 35
Figure 5-1: Bistable origami waterbomb base actuated by MAE patches .............................. 39
Figure 5-2: Relationship between elastic energy, magnetic energy, and the snap-through
field ................................................................................................................................ 41
Figure 5-3: Example of an integrated substrate and active material design ............................ 42
Figure A-1: Plot of the arch height and strain energy density of a bistable arch .................... 43
Figure A-2: Plot of the arch height and strain energy density of a nonbistable arch .............. 44
Figure B-1: Schematic of the acrylic base used for the experimental testing ......................... 45
Figure B-2: An example of SolidWorks to find the arch height. A relationship between
the 1cm scale and the SolidWorks dimensions determined the arch height. ................... 45
viii
2 LIST OF TABLES
Table 1-1: A comparison of active materials from [1] ........................................................... 5
Table 2-1: Parameter values used for COMSOL model ......................................................... 13
Table 4-1: Parametric Sweep values ...................................................................................... 30
3
ix
4 ACKNOWLEDGEMENTS
I would first like to thank my advisors Dr. Mary Frecker and Dr. Tim Simpson for their
help and guidance throughout my graduate career. I would also like to thank Dr. Paris von
Lockette for being the reader for my thesis.
There are many people here at Penn State who also helped out a lot throughout my time
here. First, I would like to thank Dr. Paris von Lockette and Rob Sheridan for all of their help
with the MAE material and modeling. I would also like to thank Saad Ahmed for taking the time
to cast PDMS for me. Finally, I would like to thank my fellow EDOG lab mates for always being
supportive and offering support whenever I needed it.
Thank you to my family and friends who have been there to cheer me on and encouraged
me to pursue my education. Thank you also to Melissa Marshall for supporting me in my career
and leadership.
x
NOMENCLATURE
Lfix Length of the fixed ends
Lpdms Length of the PDMS center
Lgap Length of the gap between MAE patches
Lmae Length of the MAE patches
Wpdms Width of the PDMS substrate
Wmae Width of the MAE patches
Tpdms Thickness of the PDMS substrate
Tmae Thickness of the MAE patches
Xdisp X-displacement
M Remanent Magnetization
Ahi Current used to create the magnetic field
Box_X X-dimension of the air box
Box_Y Y-dimension of the air box
EMAE Modulus of the MAE patches
EPDMS Modulus of the PDMS substrate
ρ Density of the MAE patches
1
1 Chapter 1
Background and Motivation
This chapter presents an introduction to the thesis and a literature review of the relevant
fields. The research objectives for the thesis are stated and an outline is also presented.
1.1 Introduction
Bistable compliant mechanisms offer the potential to be very useful in the emerging field
of origami engineering. Such devices can achieve large deformations and require only enough
input force to snap between stable positions, enabling significant shape transformations with little
energy. Throughout this thesis, the use of active materials, specifically magneto-active elastomer
(MAE) material, to generate the input force required for snap-through of the device is
investigated. This could be implemented as a part of bistable origami structures, such as the
waterbomb structure. The waterbomb, seen in Figure 1-1, is bistable and can be actuated by
applying a force to its center point. The resultant torque from correctly placed MAE patches can
also be used to generate the force required to actuate the device [1]. The use of several
waterbomb structures could be used to make more complicated structures or tessellations [2].
Figure 1-1: Example of the waterbomb
To design a bistable mechanism for origami engineering using active materials, it is
important to understand the parameters that make bistability possible. For this study, a bistable
2
arch was developed using magneto active elastomer (MAE) patches bonded to a
polydimethylsiloxane (PDMS) substrate. When the MAE material is exposed to an external
magnetic field, the patches rotate to align their internal magnetization with the external field,
generating torque that can be used to actuate the structure. Using finite element analysis (FEA)
models to understand the relationship between the thickness and modulus of the substrate and the
initial height of the arch is important to determine if the design would be bistable. A trade-space
visualization is shown to better understand the relationships and tradeoffs of several design
parameters.
This work is part of a National Science Foundation-funded EFRI-ODISSEI research
project aimed to develop multi-field origami structures using compliant mechanisms and active
materials. A visualization of this idea can be seen in Figure 1-2. This thesis seeks to bring
together these ideas with both modeling and experimentation to validate the functionality of the
MAE material and advance the FEA modeling capabilities for multiphysics simulations.
Structures like the bistable arch may one day be a part of devices incorporating other active
materials to perform a variety of tasks when different fields are applied.
Figure 1-2: A multi-field responsive origami structure actively folds from an initially flat sheet to
complex three-dimensional shapes in response to different applied fields. It is also capable of
actively unfolding from any shape back to the flat sheet.
1.2 Literature Review
This section presents related work on origami engineering, bistablity, bistable compliant
mechanisms, and the MAE material.
3
1.2.1 Origami Engineering
The use of origami principles in engineering has the potential to influence and transform
active material structures. Origami structures have been of interest to many research groups as
they can be very compact and then deploy into larger usable structures. Some researchers have
used origami modeling to decompose structures into simple geometric shapes and spherical
mechanisms [3, 4] while others use simple geometric shapes and origami structures to assemble
into something larger [2]. These larger structures can be integrated into devices and used to
change shape or orientation when in use, like a deployable solar array [5], medical devices [6], or
robot wheels [7]. For example, Lee, et al. [7] uses origami shapes to change the radius of a robot
wheel while it is in use, allowing the robot to fit under small obstacles or cover longer distances
depending on the situation. This device, which can be seen in Figure 1-3, uses motors to actuate
the deformation of the origami structures.
Figure 1-3: Deformable wheel robot [7]
4
Using active materials makes it possible to adapt these devices through the use of applied
fields. Active materials actuated by fields ranging from thermal fields to magnetic fields are
being investigated for use in origami engineering [8-12]. For example, the use of Nitinol wires, a
shape memory alloy, [10, 13] in a mesh when heated can create a variety of preprogrammed
curved shapes from an initially flat sheet. Shape memory alloys are made from nickel, titanium,
and iron, with the ability to achieve large deformations with an applied thermal field [14]. Shape
memory polymers are similar, being that they can achieve large deformations and return to the
original shape [15]. Liu, et al. [9], for instance, are using photo-thermal polymers with difference
light absorption to induce localized deformations in predetermined patterns. These polymers,
known more commonly as Shrinky-Dink, can fold in either direction, depending on where light is
applied, but once this has been done, it cannot be reversed. Photo-chemical polymers work in the
same way, but instead of reacting to the heat produced by the light, chemical bonds in the sheet
are broken by the light to induce deformation [16]. Ahmed, et al. [8] are using dielectric
elastomers to create bending and folding. Dielectric elastomers take advantage of the Maxwell
stress generated between two charged compliant electrodes around a soft elastomer film when
voltage is applied. Electrostatic forces compress the elastomer, causing an expansion through the
thickness of the film [17]. Another type of active material is Terpolymers which use
electrostrictive forces to deform, however these materials require large electric fields for any
large deformations to occur [18]. Ahmed, et al. [8] are also using MAE material to create bending
and folding for use in origami designs. MAE, which will be discussed in more detail in 1.2.4
Magneto Active Elastomer (MAE) Material, are made of hard magnetic particles mixed within an
elastomer matrix [19]. When this material is placed in a magnetic field, the patch rotates to align
with the field, generating a torque which, when part of a larger structure, can move the substrate
beneath it [1, 8, 19, 20].
All of these active materials have applications for a variety of situations. Table 1-1, from
[1], compares how these materials respond with their respective fields. MAE has a fast response
time and has the ability to move in the directions required for the arch design. This movement can
also be quickly reversed, which is an advantage for the bistable arch design. For these reasons,
MAE became the ideal choice to actuate the proposed arch.
5
Table 1-1: A comparison of active materials from [1]
Strain (%) Stress (MPa) Relative
response time
Frequency
(Hz) Bidirectional
Magneto-
active
elastomer
4-5 0.04 Fast 0-1000 Y
Dielectric
elastomer 10-200 0.1-9 Fast 0-170 N
Terpolymer 3-10 20-45 Fast 0-1000 N
Shape
memory alloy 1-8 200 Medium 0-1 N
Shape
memory
polymer
200-500 1-3 Slow 0 N
Photo-thermal
polymer 50-60 Not published Medium Non-reversible Y
Photo-
chemical
polymer
20 0.15 Slow Non-reversible Y
1.2.2 Bistablity
For a device to be bistable, it must fit several criteria. A frequently used analogy for
these devices is that of a ball on a hill, as shown in Figure 1-4. The ball begins at its first stable
position A, which is a potential energy local minimum, but when acted on by an external force, it
moves up the hill towards an unstable equilibrium position B. If just enough force is applied once
it reaches the unstable point, then it rolls on its own to the second stable equilibrium position C,
or minimum potential energy position, just like the motion of a snap-through bistable mechanism.
If there is not enough force, then it rolls back to the first position. This configuration constrains
the motion of the ball between the two stable points, just like a bistable mechanism is constrained
to move between its two stable positions [21, 22].
6
There are many advantages to using these types of bistable devices. Since the device
needs only enough force to snap, it can achieve large deformations with relatively low input
forces. In the case of the bistable arch actuated by the MAE material, the MAE patches generate
a torque when exposed to an exterior magnetic field as they try to rotate to align with the field.
This torque is what generates the input needed to get the device to snap. When the exterior
magnetic field is removed, the arch stays in the second position; however, it can be returned to
the initial position by applying an opposite field.
Figure 1-4: Ball on a hill analogy adapted from [22]. Positions A & C are stable; position B is an
unstable equilibrium position
1.2.3 Bistable Compliant Mechanisms
A compliant mechanism can transfer motion, energy, or force like traditional
mechanisms, but the unique feature that some of this mobility comes from the deflection of
flexible components, not just movable joints [23]. The use of compliant mechanisms has a variety
of advantages such as a reduction in cost, as there are usually less parts to manufacture than
traditional mechanisms, and increased performance [23]. There is a special category of these
mechanisms called bistable compliant mechanisms that move between two stable equilibrium
positions. These include latch-lock mechanisms, hinged multi-segment mechanisms, and
residual-compressive-stress buckled beam mechanisms [23-25]. These devices can be designed
as compliant mechanisms, which means that most, if not all, of the motion of the devices arise
through the deflection of flexible segments [26]. Bistable devices have two distinct positions
through which they are constrained to move, and such devices can snap from one position to the
next by storing energy during motion [22, 27]. Many devices take advantage of this such as light
switches, three ring binders, and self-closing gates [23].
7
The bistable arch investigated in this thesis is an example of a residual-compressive-
stress buckled beam. In this case, the buckled beam is fixed-fixed, meaning that both end are
constrained; however, other buckled beam designs may be pinned at the ends, or have one end
free to move [23].
Figure 1-5: Residual-Compressive-stress buckled beam in its two stable positions. Adapted from
[23].
It has been proposed that combining these mechanisms with a smart material actuator,
activation can be achieved with less power than with an electric motor [28]. Many current
methods of modeling bistable mechanisms use energy functions to determine the bistability of a
device with an input force at the center of the buckled beam [21, 22, 24, 27, 29, 30]. When the
potential energy function reaches a minimum, then the mechanism has reached a stable position.
In the case of bistable devices, there are two local minima as discussed previously. For this
thesis, two methods to determine bistablity are used: (1) the strain energy density, which shows
the local minima of the device, and (2) the removal of the magnetic field, both through FEA
analysis and through experimentation.
Many groups use the pseudo-rigid-body model to design the bistable arch [21-23, 26, 29,
31-34]. This method breaks compliant mechanisms into rigid components, such as linear and
torsional springs, dampers, and rigid beams, to approximate the motion, energy, and kinematics
of a design. This allows for quick approximations for a variety of designs without having to
develop FEA models that take a long time to run. These equivalent systems are then used to
design the material and structural parameters of each mechanism [23]. An example of the pseudo-
rigid-body model for a bistable arch and its equivalent fully-compliant mechanism is shown in
Figure 1-6.
8
Figure 1-6: Pseudo-rigid-body model of a bistable arch (a) and the equivalent fully-compliant
system (b) from [31]
1.2.4 Magneto Active Elastomer (MAE) Material
MAEs have been used in a variety of applications such as car bumper design to maximize
energy absorption in a collision [35] and to reduce noise vibration as part of a barrier system [36].
This material has also been used use in self-locomotion [37]. The self-locomotion described in
von Lockette’s work uses magnetic patches similar to those used in this thesis. As the field is
turned on, the patches of the device move to orient with the applied field. Once the field is
removed, the device flattens, resulting in a small movement forward. The device shape and
magnetic orientations can be seen in Figure 1-7.
9
Figure 1-7: MAE self-locomotive device from[37]
The MAE patches have also been used as part of an accordion bender that changes
direction based on the applied field [37]. This set up and magnetic patch orientation is similar to
that used in this thesis. The mechanism investigated in this thesis is actuated using small patches
of MAE material on each side of the arch, with the remanent magnetization orientation of the
patches pointed toward each other. In the accordion model, the remanent magnetization
orientation of the patches used also point towards one another; however, one end is free to move.
A comparison of the two models can be seen in Figure 1-8.
Figure 1-8: A. Accordion constraints and remanent magnetization orientation adapted from [37].
B. Arch constraints and remanent magnetization orientation for the work being discussed.
The MAE material considered in this thesis was fabricated using 70% of the total volume
Dow Corning HS II RTV silicone rubber compound, with 20:1 catalyst to compound ratio by
weight, mixed with 30% of the total volume 325 mesh M-type barium ferrite (BaM) particles.
10
This composite was selected as it is magnetically orthotropic. The resulting composite has an
estimated density of ρ=2800g/cm^3. The BaM materials used have a remanent magnetization of
μ0mr=0.06T [8, 19, 20, 38]. Prior to the curing process, the BaM is uniformly mixed into the
silicone rubber. Then a uniform constant field of 2T is applied to align the particles and their
magnetizations in a uniform direction and is kept throughout the curing process, giving it a
magnetization in the direction of the field [37, 39, 40]. The proposed design takes advantage of
the aligned particles, orienting the MAE patches so that they rotate to align with the field applied
in the experiment. While local gradients in the applied field will invariably produce some degree
of magnetic forces, magnetic torque is the primary actuation that drives the actuation of the
bistable device. Torque on an MAE patch can be determined from where is the
applied field, is the magnetization in the patch, and is the resulting torque. It is also
important to note that once the device has snapped to the second stable position, it can be
reversed with the application of a field in the opposite direction.
1.3 Research Objectives
The objectives for this research include the design, analysis and fabrication of bistable
compliant snap-through mechanisms using MAE actuation. This bistable arch can be used in
origami engineering to achieve large displacements with only the required force to cause snap-
through. The main research objectives are as follows:
1. Model a bistable arch through FEA software including the magnetic field, and
2. Analyze the bistable device to understand the trade-offs in performance as a function
of the design parameters, and
3. Fabricate and test a proof-of-concept device for the bistable snap-through
mechanism.
1.4 Thesis Outline
This research is focused on the design and analysis of a compliant bistable arch actuated
by MAE material.
11
In this thesis, Chapter 2 presents a detailed description of the finite element model used to
model the geometry and actuation of a bistable arch. The assumptions, study stages to simulate
the solid mechanics and magnetic field, and bistablity checks are also covered.
Chapter 3 presents the fabrication and testing of a bistable arch prototype based on the
design used in the computer model. The testing setup and procedure are described and presented
to validate the results of the model.
Chapter 4 presents a parametric study of the arch to better understand the effects of
different design parameters. The implications for design based on these results are investigated.
Chapter 5 presents a summary of the work as well as major conclusions. It also presents
contributions of the research, and states potential future work.
12
2 Chapter 2
Finite Element Model of a Bistable Arch
This chapter presents a detailed description of the finite element model used to analyze
the geometry and actuation of a bistable arch. The assumptions, study stages to simulate the solid
mechanics and magnetic field, and bistablity checks are also covered.
2.1 Design Concept
The goal in this research is to design a FEA model of a bistable arch that integrates the
solid mechanics and magnetic model capabilities of COMSOL multiphysics FEA software. The
arch design allows for large displacement and only requires just enough force to move the device
to the second stable position. The force required to move the arch is generated by torque from
MAE patches as they attempt to align with the applied field. This can help origami engineers
achieve large moments often required in their designs for folding and unfolding. COMSOL [41]
has the muliphysics capabilities required to make the bistable arch simulations possible from
building the initial shape from a flat sheet to activating and deactivating the field when needed. A
description of the model components and the results are discussed next.
2.2 FEA Model Set-up
A FEA model was developed using COMSOL [41], which is capable of coupling a
structural model with an electromagnetic model, both of which are needed to model the bistable
device. The model required two modules within COMSOL to accurately predict the motion of
the device. The first was the solid mechanics module. This was used to setup the displacement
boundary conditions of the substrate and the boundary load connection between the substrate and
the MAE patches. The second module was the magnetic module required to directly model the
effect of an applied external field to the MAE patches and to calculate the resulting Maxwell
surface stress [28]. Figure 2-1 shows the initial shape, dimensions, and boundary conditions used
for the COMSOL model. Table 2-1 summarizes the parameter values used to create the FEA
13
model in COMSOL. These values were chosen as the device made from these inputs would fit in
the magnet available for experimental testing. PDMS was used as the substrate since casts of the
material could be made to accommodate any thickness, thus the stiffness of the material could be
adjusted as needed. This material is also inert, meaning that it will not interact with the magnetic
field in any way. It is important to note that the magnetization used in the FEA model is different
than that measured in the fabricated in the material. This can arise for two reasons, the first being
that the magnetic particles may not be perfectly aligned in the matrix material, leading to a lower
overall torque possible from the material. The other reason for this can arise because of the
bonding of the magnetic particles to the matrix material. It is possible that the particles are not
perfectly bonded in the material, and this can lead to a reduction in the torque generated by the
patch. These reductions in the overall torque capabilities of the patches are reflected in the
magnetization value required for the simulations. To find the approximate value to be used in the
computer model, an average value of the magnetic field from the experimental models was found,
and then the FEA model was run to find a magnetization that matched.
Table 2-1: Parameter values used for COMSOL model
Parameter Value
Lfix 9 mm
Lpdms 42 mm
Lgap 11 mm
Lmae 7 mm
Wpdms 5 mm
Wmae 5 mm
Tpdms 1 mm
Tmae 3 mm
Xdisp 2 mm
M 0.012 T
Ahi 0.012 Wb/m (1 Wb=2.67T)
Box_X 168 mm
Box_Y 94 mm
EMAE 1.4518E6 Pa
ρ 1150 kg/m
The finite element mesh of the PDMS substrate and MAE patches uses mapped
distributions to divide the device into elements. To determine the number of elements in the
mesh, each line length (e.g. Lfix, Lpdms) was divided by five. The air box was meshed
automatically. The mesh ultimately consisted of 4,732 2D 9-node triangular elements.
14
Figure 2-1: Initial shape, dimensions, and boundary conditions for COMSOL model
2.2.1 Solid Mechanics Module
The PDMS substrate is modeled as a hyper-elastic material using the Mooney-Rivlin
two-term approximation [42]. The C10 and C01 values were found empirically to be 63 kPa and 31
kPa, respectively [43]. The MAE material is modeled through the Magnetic Field module built
into COMSOL [41]. The magnetic forces and moments acting on the MAE patches are resolved
into a boundary load acting on each of the MAE domains’ boundaries. The boundary load is
calculated using the Maxwell surface stress tensor defined in the Magnetic Field module. The
magnetization M is entered as a vector quantity in COMSOL whose direction alternates in each
MAE patch. Since the patch on the left of the model in Figure 2-1 has a magnetization that points
in the positive x direction, the remanent magnetization is entered as positive, and vice versa for
the other patch. The direction of this magnetization must move with the patch as it moves. This
is done by forcing the remanent magnetization to move spatially with the patches within
COMSOL, i.e., the remanent magnetization is an Eulerian quantity.
15
2.2.2 Magnetic Material Module
The magnetic field is created using an air box that simulates the two faces of the C-
magnet. This is again developed using the Magnetic Field module in COMSOL. The top and
bottom sections of the box are modeled as perfect magnetic conductors. The vector valued
solution variable 𝐴 is used to set the magnetic boundary conditions, 𝐴 0 on the left boundary
and 𝐴 𝐴ℎ𝑖 on right. By varying the magnitude of 𝐴ℎ𝑖 the relationship between Ahi and the
resulting solution for magnetic field as generated by COMSOL can be seen in Figure 2-2. The
size of the air box had to be sufficiently large to ensure that there was convergence around the
field required to generate snap. To do this, the size of the box was increased until the magnetic
field value required for snap converged to the same value, showing that the box size was
sufficiently large to not influence the required field. Figure 2-3 shows the Maxwell surface
stresses generated as the field is applied. This surface stress generates the effective torque of the
MAE patches and ultimate snap-through of the device.
Figure 2-2: Relationship between Ahi & magnetic field
16
Figure 2-3: Maxwell surface stress on the boundaries of the MAE patches
2.2.3 Study Steps
The model executes in three time-dependent steps. The first step uses only the solid
mechanics parts of the model, and the magnetic field is kept off, as shown in Figure 2-7 steps A
and B. This creates the initial shape of the arch. In Figure 2-7B, the left and right ends are given
prescribed displacements toward one another in the x direction, but they are fixed in the y
direction. Both ends are displaced toward one another to keep the mechanism centered in the air
box used to create the magnetic field. To ensure that the arch develops the initially curved shape,
the initial displacement is incremented linearly with time, while simultaneously solving for the
internal stress state before reaching the final position. A plot showing the displacement
incremented with time can be seen in Figure 2-4. Since this internal stress resulting from the
initial displacement is an important part of the design and functionality of bistable devices, it is
important to model the fabricated mechanisms from an initially flat position to ensure that the
FEA model accounted for internal stress.
17
Figure 2-4: The initial displacement increases linearly with time
The second step of the analysis uses the final solution of the first step as its initial
condition, as seen in Figure 2-7 steps C through E. It takes the initially curved shape and applies
a magnetic field, modeled as Ahi, which is again slowly incremented with time. This increase can
be seen in Figure 2-5. The value for the magnetic field at a given time is found using the
relationship between Ahi and the magnetic field shown in Figure 2-2. The magnetic field is
directed from the top to the bottom of the magnet, causing the MAE patches to rotate and induces
a torque. This torque slowly increases with the increasing field, eventually generating a moment
large enough to cause the device to snap from its first stable position.
18
Figure 2-5: Ahi increases linearly with time
The third and final step of the simulation is the removal of the magnetic field. This is
achieved by quickly ramping down the field generated in the second step to zero. A plot of this
can be seen in Figure 2-6. If this ramp down is done too quickly, then the solver has trouble
finding a solution; so, it is important to spread this step out over a few time increments.
Figure 2-6: Ahi scalar as a function of time during the third study step
19
2.2.4 Determining Bistablity
To determine if the configuration of the device is bistable, the magnetic field is removed
during the third step, shown in the final image in Figure 2-7. If the device remains in the second
position, then it is bistable. If it returns to the initial arch shape, then the device is not bistable.
This is also shown by the strain energy density of the design (see Figure 2-8). When the device is
bistable, the strain energy density reaches a local minimum. However, if the device is not
bistable, then there will not be a local minimum at the snap point, and the device will return to the
first stable state when the field is removed. The design on the left in Figure 2-7 is bistable,
whereas the design on the right snaps through but is not bistable. The strain energy densities for
both cases are shown in Figure 2-8 and labelled corresponding to the steps in Figure 2-7. A plot
of the arch height and the strain energy density together can be found in Appendix A.
Figure 2-7: Steps used in the COMSOL model, 1 is bistable and 2 is nonbistable. A is the initial
shape, B is the initial stable arch shape, C is the initial application of the field, D is the
displacement as the field is applied, E is the position after the snap, and F is the second stable
position after the field is removed.
20
Figure 2-8: Strain energy density of a bistable (top plot) and non-bistable (bottom plot) design
with corresponding steps shown in Figure 2-7
2.3 Discussion
The FEA model behaved as anticipated. As the field was applied, the arch began to
deform and eventually reached the snap-through point when it moved from the first stable
position to the second. The FEA model also showed that the device was bistable when the field
was removed and the arch remained in the second position. This was confirmed in physical
experiments using the device, discussed in detail in Chapter 3.
Figure 2-9 shows the y displacement of the center point height as the magnetic field
increases from the simulations. Once the magnetic field reached 0.069 T, then the MAE patch
generated enough torque to cause the device to snap. This is shown by the instantaneous decrease
in arch height as it snaps from the first position to its second.
21
Figure 2-9: FEA prediction of center point displacement as the magnetic field increased
22
3 Chapter 3
Experimental Validation of the FEA Model
This chapter presents a validation of the COMSOL model using hand-fabricated
experimental prototypes. The test set-up, procedure, and results are discussed in detail.
3.1 Experimental Validation
An experimental test setup was developed to determine the validity of the FEA models
using the dimensions of a design that is predicted by the FEA model to be bistable (see Table
2-1). The experimental set-up developed to ensure that there would be no interference with any
metal parts surrounding the test fixture, and that each test would be lined up properly. The
magnetic field required to cause the device to snap is compared to that predicted by the FEA
model.
3.1.1 Experimental Set-up
The experiment was set-up using the same substrate and MAE material used in the FEA
model. The PDMS strips used in the experiment were cut from a larger sample of the material
with a thickness of 1 mm. The MAE patches were all cut from a long strip that was 5 mm wide.
There were three test samples were assembled and attached to acrylic bases using PermaBond
268. The acrylic bases were laser cut with a 38cm long inner rectangular opening to ensure the
proper initial height and displacement. A schematic of the base can be found in Appendix B.
Prior to attachment to the acrylic base, the MAE patches were attached to the PDMS
substrate using the same glue. The surface of the base and the PDMS substrate had to be sanded
to ensure a good bond with the glue. These were mounted in place in the magnet by a wood
stand, since wood is not magnetic. Figure 3-1 shows a schematic of the test fixture. The DC
regulated power supply, CSI3020X, had a voltage display which was used to calculate the
magnetic field.
23
To generate the magnetic field, an electromagnet with an iron core and a 3.9 Ω resistance
was used. The electromagnet was constructed in the shape of a “C” with two facing poles (also
known as a c-magnet) and was powered by the DC CSI3020X power source. Use of the power
source allowed a controlled voltage to be applied to the c-magnet; however, for further analysis
and relationships the magnetic field generated between the two facing poles needed to be
quantified. Therefore, to quantify the relationship between the applied voltage and the magnetic
field strength, a LakeShore 475 DSP Gauss meter was used to measure the field strength in Tesla
units. The probe of the Gauss meter was held in the center between the two facing poles of the c-
magnetic, and measurements of the magnetic field strength were taken at each whole interval
from 0 Volts to 30 Volts. To determine the center between the poles, the pole faces were
measured and marked for better precision. The electromagnet was also powered for the duration
of the 0 to 30 Volt measurements, and magnetic field strengths were recorded once they reached a
steady value. As expected, with increasing voltage, the magnetic field between the two pole faces
also increased as shown in Figure 3-2. This relationship was used to calibrate the voltage values
displayed during the experiments with magnetic field values.
Figure 3-1: Test setup for experimental validation
24
Figure 3-2: Magnetic field as the voltage increases on the power supply
For the experiment, each sample was placed in its initial arch shape on a wood stand.
The wood was cut so that the samples would be level and centered vertically in the magnet.
There were guidelines drawn on to the top of the wood blocks to ensure that the samples were
centered horizontally and into the plane. This setup can be seen in Figure 3-3. Slowly, the
voltage on the power supply was increased until the device snapped and settled into its second
stable position. Then, the field was removed to ensure that the device was bistable.
Each of the three samples did snap through, and all of them were bistable, i.e., they
stayed in the snap-through position when the magnetic field was removed. Each test started at 0
Tesla, and the magnetic field was slowly increased as it was in the FEA model. A video recording
was made using a Canon EOS 7D camera with a fixed focal length lenses and 18 megapixel
resolution. The camera recorded the shape of the arch as well as the voltage displayed by the
power supply. A ruler was placed near the arch to be used to find the height of the center point
using SolidWorks. An example of how SolidWorks was used to find the arch height can be found
in Appendix B.
25
Figure 3-3: Experimental test setup with guidelines on the wood stand
Figure 3-4 shows the three arches that were constructed for testing, and Figure 3-5 shows
a screenshot of the information gathered from the camera. The height data was used to create a
plot of the center point displacement as the mechanism moved from the first stable position to the
second with the application of a magnetic field.
Figure 3-4: Three samples for experimental validation
26
Figure 3-5: Image from the camera indicating the voltage and arch height
3.1.2 Experimental Results
Each sample was constructed and then tested five times to provide a set of 15 runs.
Between each run, the samples were manually reset to the initial arch shape. The five test run
heights were then averaged to find the trend for each of the samples. The plot of the average
center point heights for each fixture can be seen in Figure 3-6. The results from the FEA model
can also be seen in this figure. As seen in this figure, little torque is generated by the MAE
patches at low voltages, and there is little to no movement in the samples. All of them reached a
point when the torque generated by the MAE patches caused the devices to snap just as it did in
the FEA model. When the field was removed, the samples stayed in the second stable position.
Error bars are shown with a 95% confidence level, demonstrating that the results for each sample
are repeatable. The magnetic field required for snap-through of the three samples range from
.065T to .072T. The initial arch height for the three samples ranges from 7.1mm to 8.0mm. This
shows that there is some variation in the fabrication of each device, specifically the initial
displacement of the fixed ends. There is also a slight increase in the center arch height just before
the samples reached the snap-through point. This could arise when the MAE patches are not
perfectly symmetric, and one side pushes the other up slightly. This can be seen in Figure 3-7B.
27
Figure 3-6: Plot of the center point height for the three test fixtures. The FEA model results are
also shown.
28
Figure 3-7: Stages of snap-through as the device is exposed to an increasing magnetic field
Figure 3-7 shows the motion of the first sample as it was exposed to the magnetic field.
When there was no field or very little field, then the device did not show any movement. As the
field grew stronger, the device began to elastically deform and ultimately snapped through to
settle in the second stable position. When the magnetic field was turned off, the device remained
in the second position.
3.2 Discussion
There are several factors that may contribute to the visible differences between the FEA
model and the experimental results. To begin, each of the experimental samples were constructed
by hand. This could lead to imperfections in the size, shape, and initial displacement of the
29
devices. This can be seen in Sample 1, which consistently has a higher arch height that the other
samples created. This means that the initial displacement of this design is slightly offset from the
other prototypes. We have found that very slight differences in the dimensions of the samples can
lead to differences in the magnetic field required to get the devices to snap. It was also observed
that the left and right hand sides of the arch do not move perfectly symmetrically as expected, as
shown in Figure 3-7B. This can cause the sample to have a somewhat different snap-through
point than predicted by the FEA model.
Another contributing factor could be the remanent magnetization of the MAE patches.
This difference can be seen in the snap-field values for the three prototypes. If the magnetic
particles are not uniformly distributed throughout the larger sample during the curing process
used to construct the patches, and aligned similarly in each patch, then there can be slight
differences between the magnetization of the patches. For this reason, it is important to
accurately measure this variation in the MAE material using, for example, x-ray diffraction and
vibrating sample magnetometry. It is also possible that some particles may be able to rotate
freely inside the silicone rubber, without contributing to the overall torque. This would result in a
lower remanent magnetization than expected.
30
4 Chapter 4
Parametric Variation of Bistable Arch
This chapter details the parametric sweep set-up to evaluate the results from the study.
The design implications are also discussed.
4.1 Parametric Sweep Set-up
It is important to understand the way various parameters in the model affect the magnetic
field required to get the bistable arch to snap and the bistablity of the device. To investigate this,
several parameters were varied using the FEA model developed in Chapter 2: PDMS substrate
thickness (TPDMS), initial displacement (Xdisp), MAE length (LMAE), and the length of the gap
between the MAE patches (Lgap). The values for the parameters can be found in
Table 4-1.
All combinations of these variables were used to generate a total of 225 designs. The initial and
final arch heights were recorded to find the total displacement of the arch along with the magnetic
field at snap-through, and the maximum von Mises stress. The bistability of the design was also
noted based on the strain energy density throughout the simulation. The results from this study
were analyzed using a trade space visualization tool known as ATSV. This software program
was developed by the Applied Research Lab at Penn State to give users the ability to intuitively
visualize multi-dimensional trade spaces and derive relationships between design parameters[44].
Table 4-1: Parametric Sweep values
Parameter Sweep Values (mm)
TPDMS PDMS Thickness 1, 1.25, 1.5, 1.75, 2
Xdisp Initial Displacement 1, 2, 3, 4, 5
LMAE MAE Length 5, 7, 9
Lgap Length MAE Gap 9, 11, 13
31
4.2 Parametric Sweep Results
The results of the parametric sweep showed many design implications. The full results
from the study can be found in Appendix C. The initial displacement, PDMS thickness, magnetic
field, von Mises stress, MAE separation, and MAE length were normalized with respect to the
maximum and minimum values of the variable shown in Equation 4-1.
𝑖
𝑖 = Rel_Value (4-1)
The parametric sweep revealed a relationship between the thickness of the PDMS
substrate and the magnetic field required to cause snap-through. As the design became thicker,
the magnetic field required for snap-though increased. There is also a relationship between the
initial displacement of the device and the magnetic field. Again, as the initial displacement
increased, the magnetic field required increased. These relationships can be seen in Figure 4-1.
The bistablity of the device is also strongly dependent on the thickness and initial displacement of
the PDMS substrate. As the substrate designs got thicker and the initial displacement became
smaller, the designs tended to not be bistable. In Figure 4-1, the bistable designs are shown in
blue, whereas the non-bistable designs are shown in yellow. While the non-bistable did not stay
in the second position, the magnetic patches did generate the torque required to move the designs
to a second position.
32
Figure 4-1: A glyph plot in ATSV of the relationship between the thickness and initial
displacement of the PDMS substrate and the magnetic field required to cause snap-through of the
device design
The length of the MAE patches and the location of the patches on the MAE substrate had
no effect on the bistablity of the device. There was, however, an effect on the magnetic field
required to actuate the device. Figure 4-2 shows an glyph plot in ATSV of the normalized
thickness and displacement with varying MAE patch sizes to represent the different lengths of
MAE patches simulated, the larger the block, the larger the patch. This shows that as the MAE
patch size increased, the less magnetic field was needed to actuate the device.
33
Figure 4-2: A glyph plot in ATSV showing the normalized MAE length variation and its effect on
the magnetic field required for snap-though
There was more variation in the separation between the MAE patches and the field
required to actuate the device. Figure 4-3 shows a glyph plot in ATSV plot of the normalized
thickness and displacement with varying block sizes to represent the different separations of
MAE patches. The larger the block, the more separation there was between the patches. In some
cases, as patch separation decreased, the lower the field strength required to cause snap-through.
However, this was not always the case. More studies investigating the ideal placement of the
MAE patches is required to determine any trends with this parameter.
34
Figure 4-3: A glyph plot in ATSV showing the normalized MAE separation variation and its
effect on the magnetic field required for snap-though
4.3 Bistable Region
Additional simulations were performed to refine the region of bistability. This region can
be seen in Figure 4-4. As stated previously, designs that have greater thickness with a lower
initial displacement tended to not be bistable. For this particular substrate design, with 5mm
width and 42mm length, the region of bistablity occurs when the initial displacement is around
75% of the PDMS thickness. At this initial displacement, the arch height was still sufficiently
low, and did not create a bistable design. As the designs are displaced, the strain energy density in
the design increases linearly. This could lead to the linear trend seen from these sweeps. When
this was repeated for a substrate of 35mm, the ratio did not hold up. For this design, the bistable
region began around 60% of the PDMS thickness. More studies are needed to understand how
35
this region is influenced by the initial length and width of the PDMS substrate.
Figure 4-4: Bistable region for the PDMS substrate design
4.4 Parametric Sweep Discussion
The results show that using a thinner substrate leads to lower required actuation field.
This has limits, however, as using a substrate that is too thin may not be able to support the
weight of the MAE patches. Using larger MAE patches can also reduce the required field with the
tradeoff of using more material, making the devices more expensive to manufacture. More studies
are needed to understand the ideal locations of the patches and select the ideal MAE patch size
for a specific arch height. If only a little movement of the arch is required for the design, then it is
important to use a design that is low enough, yet still bistable. The force generated by the arch
36
throughout its motion is important to understand. Designs with thicker substrates and greater
initial displacements require more torque to actuate but also exert more force. This information
helps designers select the appropriate thickness and initial displacement to achieve specific design
goals.
37
5 Chapter 5
Conclusions and Recommendations
In this chapter a summary of the thesis, extensions to other designs, and future work for
this area are presented.
5.1 Summary and Conclusions
The modeling and experimental work in this thesis has the potential to impact the field of
origami engineering. Achieving large deformations through the use of bistable devices has been
demonstrated with the use of active materials. Specifically, MAE patches can generate the torque
required to actuate an arch design. An FEA model, using a multiphysics software package, was
developed to predict the bistability of an arch design and determine the magnetic field required to
get the device to snap from its first to second stable position. This required the use of a time
dependent study to execute the different steps throughout the development and actuation of the
arch. Experimental results validated this model and illustrated the importance of precision in
crafting the devices. Understanding the relationship between design parameters is important to
ensure the bistability of the arch. Using trade-space visualization, it is possible to understand the
trade-offs between larger displacements and a greater field required to make those displacements
possible as well as the region where bistablity occurs.
The development of the FEA models showed the importance of developing the correct
mesh size. If the mesh of the air box was too coarse, the magnetic field required to snap the arch
would not be reflected correctly in the solution. This is also true for the arch itself. If the mesh
was too coarse, the solution for the strain energy density was not correctly reflected, and the mesh
had to be adjusted. It is also important not to make the mesh too fine, as this leads to a large
increase in the time it takes to run each simulation. These studies also showed the sensitivity of
the solvers to the large deformations in the simulations. Often times, if the time steps for each
study were too large, the solution would not converge when the large deformation should have
occurred. While this meant that the simulations ran for long periods of time, it was necessary to
find a solution which converged.
38
Developing the experimental test set-up required very precise alignment of the
prototypes. It was also important to use a stand that would not interfere with the magnetic field.
Since most clamps are metal, wood blocks were required to make a stand that would maintain a
constant height. Once the proper height was established, the alignment of the prototypes in the
magnet had to be fixed. Consistency between each of the runs was critically important to ensure
that the results from the test runs were accurate and comparable. Building the prototypes is a
delicate process. The pieces are small, so the slightest misalignment can cause large discrepancies
between seemingly identical models. This is true of the development of the MAE patches as well.
If the mixture is not thoroughly mixed with the proper amount of magnetic material, there can be
inconsistencies in each sample.
Running the parametric sweep required a lot of patience. Ideally, the model could be run
in batch mode so hundreds of combinations could be done without having to manipulate the
model for each new design. With this design, with a large deformation occurring at different
times for each model, occasionally, a design would not converge and the batch simulation would
fail. This meant that batch mode could not be consistently used, as it is impossible to predict
when a particular design would not converge. In the future, it could be possible to write a script
that could adjust the time steps and restart the simulations to try to overcome this issue.
5.2 Extension to Origami Engineering: A MAE-actuated Waterbomb
There are many examples of bistable structures in traditional origami. For example, the
waterbomb is bistable and can be actuated using a force applied to the center point or by active
materials such as MAE. The use of several waterbomb structures could be used to make more
complicated structures [2]. An example can be seen in Figure 5-1, where MAE patches have been
bonded to a creased paper substrate. The structure begins in one stable position, and when a
magnetic field is applied, the torque generated by the four MAE patches causes snap-through to
the second stable position. When the magnetic field is removed, the waterbomb remains in the
second stable position. The process can then be reversed by reversing the direction of the
magnetic field. The bistable arch developed in this paper is similar in nature to the waterbomb
structure. Bowen, et al. [1] are working to develop a dynamic model of the waterbomb to better
understand the placement and alignment of the MAE patches to activate bistable behavior. With
this knowledge, they plan to develop a waterbomb using a PDMS molded base and MAE patches
39
based on the dynamic model orientations. This can be used to validate the results found and
optimize the magnetic orientation of the MAE patches. To move to a finite element model, it is
important to understand how the boundary conditions of the waterbomb need to be applied. In the
dynamic model [1], one of the waterbomb panels is held fixed, while the others rotate around it.
This is one possible method to use when moving towards an FEA model. It may also be possible
to model the actuation of the waterbomb on a rigid surface, like a table, by constraining the center
point to move only up and down. The valley folds in position 1 of Figure 5-1 would require
sliding constrains as the waterbomb transitions between the two states.
Figure 5-1: Bistable origami waterbomb base actuated by MAE patches
5.3 Future Work
Future work is required to design and optimize the model to maximize the height of the
center point while minimizing the magnetic field required to achieve snap-through of the device.
This requires more studies to identify the relationship between the size of the MAE patches and
the separation between them on the PDMS substrate. These devices could be used as a switch to
toggle a device without the need for mechanical manipulation. Instead, applying a field around
the device can cause the switch to move from one position to the other. It is also important to
perfect the manufacturing technique to improve the design of the mechanisms to ensure that the
predicted field is sufficient to actuate the device. It is also important to better understand the
behavior of the BaM particles and how they interact with the silicone rubber when a magnetic
field is applied. This understanding can help predict the magnetization value required to model
devices using COMSOL.
40
The FEA simulations for this study took between 5 and 15 minutes to run. Developing a
relationship between the buckling stiffness and the torque generated by the magnets could be used
to predict the magnetic field required for snap-through. This would eliminate the need for many
simulations and allow for more design iterations. It is also important to develop an analytical
relationship between the thickness, initial displacement, and bistability for a range of substrate
lengths. With this understanding, and an understanding of the buckling stiffness, it may be
possible to predict the bistablity of a design without running FEA simulations. The interaction
between the magnets may come into consideration as well. There could be a point where the
interaction between these magnets may be strong enough to overcome the non-bistablity of a
design. Preliminary investigations of the ratio between the elastic energy and magnetic energy
correlated with the snap-through field can be seen below in Figure 5-2. The elastic energy is
calculated by the modulus of the substrate, area moment of inertia, and substrate length, seen in
Equation 5-1. The magnetic energy is a function of the magnetization and volume of the MAE
patch, seen in Equation 5-2. This plot shows that there is a region where the snap-through field is
low with a high energy ratio. Further analysis of this region can help identify bistable designs
without the need for time consuming simulations.
𝐸𝑙𝑎𝑠𝑡𝑖𝑐 𝐸𝑛𝑒𝑟𝑔𝑦 𝐸 𝐼
𝐿 (5-1)
𝑎𝑔𝑛𝑒𝑡𝑖𝑐 𝐸𝑛𝑒𝑟𝑔𝑦 𝐿𝑀𝐴𝐸𝑊𝑀𝐴𝐸 𝑀𝐴𝐸 (5-2)
41
Figure 5-2: Relationship between elastic energy, magnetic energy, and the snap-through field
Using principles from robust design can reduce the variation in experimental prototypes.
Robust design seeks to reduce the effects of variation by minimizing the effects of controllable
and uncontrollable variables [44]. For example, ensuring that the MAE patches all come from the
same batch can help reduce uncertainties in the magnetization of the patch and reduce any
environmental effects between samples, as the patches will all have the same manufacturing
process. By applying these ideas, variation in MAE magnetization, PDMS thickness, and
prototype development can be reduced to generate more accurate designs and results.
Another area that requires investigation is the force generated as the arch moves from the
first to the second stable position. When the arch is used as an actuator, it could be used to push
another piece of a larger design. It could also be used to move a larger part of the structure with
the momentum generated through snapping. Understanding the force requirements of the design
and the capabilities of the arch is critical to making this type of actuation possible. Designs with
42
thicker bases and greater initial heights require more torque to actuate and would likely have
more force as they snap through. This would, however, require more energy input to make this
happen. There is also a limit to the initial displacement as too much can cause interference
between the two sides.
In the future, the MAE patches could be cast as part of a monolithic sheet to eliminate the
need to glue patches to the substrate, an example of which can be seen in Figure 5-3. This would
reduce potential errors in the development and alignment of patches that may slip while being
attached. With this, it is possible to integrate other active materials into the sheet to allow for
multi-field active origami structures. This will require coupling between several external fields,
which presents a challenge for finite element modeling experimental designs. The integration of
these MAE patches into a substrate can affect the stiffness of the arch or other design altering the
magnetic field strength required for snap-through to occur. This must be taken into account when
adjusting FEA models to accommodate the new material.
Figure 5-3: Example of an integrated substrate and active material design [45]
5.4 Acknowledgments
I gratefully acknowledge the support of the National Science Foundation EFRI grant
number 1240459 and the Air Force Office of Scientific Research. Any opinions, findings, and
conclusions or recommendations expressed in this material are those of the author and do not
necessarily reflect the views of the National Science Foundation.
43
Appendix A: Complete results for displacement and strain energy density
Figure A-1: Plot of the arch height and strain energy density of a bistable arch
44
Figure A-2: Plot of the arch height and strain energy density of a nonbistable arch
45
Appendix B: Extra images for experimental model
Figure B-1: Schematic of the acrylic base used for the experimental testing
Figure B-2: An example of SolidWorks to find the arch height. A relationship between the 1cm
scale and the SolidWorks dimensions determined the arch height.
46
Appendix C: Complete results for the parametric Sweep
Thickness
Initial
Displacement
MAE
length
MAE
separation
Peak
height
Valley
Height
Total
displacement snap field
VM stress (@
flip)
Bistable
(y/n) torque (Nm)
1 1 5 13 3.96 -3.96 7.92 0.009833925 28000 y 0.000418386
1 3 5 13 6.6 -6.6 13.2 0.020105438 50900 y 0.000855389
1 5 5 13 7.95 -7.95 15.9 0.028144013 67000 y 0.001197392
1 1 7 13 3.96 -3.96 7.92 0.00894075 25200 y 0.000380386
1 3 7 13 6.6 -6.6 13.2 0.0178725 42500 y 0.000760389
1 5 7 13 7.95 -7.95 15.9 0.0250179 56100 y 0.001064391
1 1 9 13 3.96 -3.96 7.92 0.007600988 23900 y 0.000323385
1 3 9 13 6.6 -6.6 13.2 0.014746388 41600 y 0.000627388
1 5 9 13 7.95 -7.95 15.9 0.01965885 51200 y 0.000836389
1 1 5 11 3.96 -3.96 7.92 0.00894075 29000 y 0.000380386
1 3 5 11 6.6 -6.6 13.2 0.0178725 52200 y 0.000760389
1 5 5 11 7.95 -7.95 15.9 0.024571313 68500 y 0.001045391
1 1 7 11 3.96 -3.96 7.92 0.009387338 30000 y 0.000399386
1 3 7 11 6.6 -6.6 13.2 0.020105438 57200 y 0.000855389
1 5 7 11 7.95 -7.95 15.9 0.028144013 74700 y 0.001197392
1 1 9 11 3.96 -3.96 7.92 0.009387338 27600 y 0.000399386
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1 5 5 9 7.95 -7.95 15.9 0.020105438 64900 y 0.000855389
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48
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50
1.25 5 5 9 7.95 -7.95 15.9 0.0106625 95892.75035 y 0.000453638
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51
1.75 5 5 9 7.95 -7.95 15.9 0.0644625 282429.9014 y 0.002742568
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52
2 2 7 13 5.44125 -5.44125 10.8825 0.0694 51590 y 0.002952635
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53
1.5 4 9 11 7.43625 -7.43625 14.8725 0.0518 88050 y 0.00220384
1.5 2 5 9 5.44125 -5.44125 10.8825 0.0392 65112.5 y 0.001667771
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1.25 4 9 13 6.6 -6.6 14.8725 0.0323 57274.2 y 0.001374209
1.25 2 5 11 3.96 -3.96 10.8825 0.0244 55285.8 y 0.001038102
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1.25 2 7 11 3.96 -3.96 10.8825 0.0259 47705.18 y 0.00110192
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1.25 4 5 9 6.6 -6.6 14.8725 0.0238 82898.44 y 0.001012575
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1.25 4 9 9 6.6 -6.6 14.8725 0.0339 132296 y 0.001442281
1.75 2 5 13 3.96 -3.96 10.8825 0.0609 58698.6 y 0.002591001
1.75 4 5 13 6.6 -6.6 14.8725 0.1061 155337.6 y 0.004514043
54
1.75 2 7 13 3.96 -3.96 10.8825 0.0529 53247.2 y 0.00225064
1.75 4 7 13 6.6 -6.6 14.8725 0.0925 107065.2 y 0.003935429
1.75 2 9 13 3.96 -3.96 10.8825 0.046 55059.6 y 0.001957078
1.75 4 9 13 6.6 -6.6 14.8725 0.0796 85508.4 y 0.003386596
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1.75 4 7 11 6.6 -6.6 14.8725 0.0999 227510 y 0.004250263
1.75 2 9 11 3.96 -3.96 10.8825 0.0565 54995.2 y 0.002403802
1.75 4 9 11 6.6 -6.6 14.8725 0.0839 131443.2 y 0.00356954
1.75 2 5 9 3.96 -3.96 10.8825 0.0538 61213.2 y 0.00228893
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1.75 2 7 9 3.96 -3.96 10.8825 0.0505 52843 y 0.002148531
1.75 4 7 9 6.6 -6.6 14.8725 0.0729 241787 y 0.003101543
1.75 2 9 9 3.96 -3.96 10.8825 0.0576 76361.82405 y 0.002450602
1.75 4 9 9 6.6 -6.6 14.8725 0.0874 385554.6265 y 0.003718448
55
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