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

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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

1 3 9 11 6.6 -6.6 13.2 0.019212263 49500 y 0.000817389

1 5 9 11 7.95 -7.95 15.9 0.026357663 63200 y 0.001121391

1 1 5 9 3.96 -3.96 7.92 0.007600988 27400 y 0.000323385

1 3 5 9 6.6 -6.6 13.2 0.015192975 52400 y 0.000646388

47

1 5 5 9 7.95 -7.95 15.9 0.020105438 64900 y 0.000855389

1 1 7 9 3.96 -3.96 7.92 0.008047575 31100 y 0.000342385

1 3 7 9 6.6 -6.6 13.2 0.01608615 55700 y 0.000684388

1 5 7 9 7.95 -7.95 15.9 0.022338375 74500 y 0.00095039

1 1 9 9 3.96 -3.96 7.92 0.009833925 35900 y 0.000418386

1 3 9 9 6.6 -6.6 13.2 0.0214452 68500 y 0.00091239

1 5 9 9 7.95 -7.95 15.9 0.03037695 94300 y 0.001292393

2 1 5 13 3.96 -3.96 7.92 0.033503063 35900 n 0.001425394

2 3 5 13 6.6 -6.6 13.2 0.119843313 120000 y 0.005098755

2 5 5 13 7.95 -7.95 15.9 0.16256685 412000 y 0.006916435

2 1 7 13 3.96 -3.96 7.92 0.02680425 35100 n 0.001140392

2 3 7 13 6.6 -6.6 13.2 0.102724125 88400 y 0.004370416

2 5 7 13 7.95 -7.95 15.9 0.14113065 223000 y 0.006004428

2 1 9 13 3.96 -3.96 7.92 0.022338375 36700 n 0.00095039

2 3 9 13 6.6 -6.6 13.2 0.08634925 82100 y 0.003673744

2 5 9 13 7.95 -7.95 15.9 0.120587625 135000 y 0.005130422

2 1 5 11 3.96 -3.96 7.92 0.032609888 40200 n 0.001387393

2 3 5 11 6.6 -6.6 13.2 0.112400188 144000 y 0.004782086

2 5 5 11 7.95 -7.95 15.9 0.14827605 626000 y 0.006308431

2 1 7 11 3.96 -3.96 7.92 0.025911075 35900 n 0.001102391

2 3 7 11 6.6 -6.6 13.2 0.109422938 155000 y 0.004655418

2 5 7 11 7.95 -7.95 15.9 0.1500624 809000 y 0.006384431

2 1 9 11 3.96 -3.96 7.92 0.021891788 35600 n 0.00093139

2 3 9 11 6.6 -6.6 13.2 0.10421275 97100 y 0.00443375

2 5 9 11 7.95 -7.95 15.9 0.147382875 327000 y 0.00627043

2 1 5 9 3.96 -3.96 7.92 0.031716713 34300 n 0.001349393

2 3 5 9 6.6 -6.6 13.2 0.099002563 139000 y 0.004212081

48

2 5 5 9 7.95 -7.95 15.9 0.127733025 563000 y 0.005434424

2 1 7 9 3.96 -3.96 7.92 0.0250179 36500 n 0.001064391

2 3 7 9 6.6 -6.6 13.2 0.093792375 160000 y 0.003990413

2 5 7 9 7.95 -7.95 15.9 0.1286262 739000 y 0.005472424

2 1 9 9 3.96 -3.96 7.92 0.0214452 35600 n 0.00091239

2 3 9 9 6.6 -6.6 13.2 0.104957063 184000 y 0.004465417

2 5 9 9 7.95 -7.95 15.9 0.1500624 1040000 y 0.006384431

1.5 1 5 13 3.96 -3.96 7.92 0.025911075 37000 n 0.001102391

1.5 3 5 13 6.6 -6.6 13.2 0.060298313 82500 y 0.002565402

1.5 5 5 13 7.95 -7.95 15.9 0.084116313 107000 y 0.003578743

1.5 1 7 13 3.96 -3.96 7.92 0.021891788 34500 n 0.00093139

1.5 3 7 13 6.6 -6.6 13.2 0.052706325 64900 y 0.0022424

1.5 5 7 13 7.95 -7.95 15.9 0.073695938 83900 y 0.003135407

1.5 1 9 13 3.96 -3.96 7.92 0.018319088 35100 n 0.000779389

1.5 3 9 13 6.6 -6.6 13.2 0.043327988 60900 y 0.001843397

1.5 5 9 13 7.95 -7.95 15.9 0.061786938 65100 y 0.002628736

1.5 1 5 11 3.96 -3.96 7.92 0.022784963 37200 n 0.00096939

1.5 3 5 11 6.6 -6.6 13.2 0.054939263 96400 y 0.002337401

1.5 5 5 11 7.95 -7.95 15.9 0.075928875 141000 y 0.003230407

1.5 1 7 11 3.96 -3.96 7.92 0.0214452 38100 n 0.00091239

1.5 3 7 11 6.6 -6.6 13.2 0.057618788 83300 y 0.002451401

1.5 5 7 11 7.95 -7.95 15.9 0.078906125 168000 y 0.003357075

1.5 1 9 11 3.96 -3.96 7.92 0.020998613 37500 n 0.00089339

1.5 3 9 11 6.6 -6.6 13.2 0.055832438 74500 y 0.002375401

1.5 5 9 11 7.95 -7.95 15.9 0.034842825 98300 y 0.001482394

1.5 1 5 9 3.96 -3.96 7.92 0.020552025 38500 n 0.00087439

1.5 3 5 9 6.6 -6.6 13.2 0.0464541 87900 y 0.001976398

49

1.5 5 5 9 7.95 -7.95 15.9 0.025464488 122000 y 0.001083391

1.5 1 7 9 3.96 -3.96 7.92 0.018765675 38200 n 0.000798389

1.5 3 7 9 6.6 -6.6 13.2 0.047793863 91300 y 0.002033398

1.5 5 7 9 7.95 -7.95 15.9 0.02680425 230000 y 0.001140392

1.5 1 9 9 3.96 -3.96 7.92 0.019212263 38400 n 0.000817389

1.5 3 9 9 6.6 -6.6 13.2 0.05895855 101000 y 0.002508402

1.5 5 9 9 7.95 -7.95 15.9 0.036629175 477000 y 0.001558395

1.25 1 5 13 3.96 -3.96 7.92 0.017649206 31100 y 0.000750889

1.25 3 5 13 6.6 -6.6 13.2 0.038192231 62600 y 0.001624895

1.25 5 5 13 7.95 -7.95 15.9 0.0534125 91109.57783 y 0.002272444

1.25 1 7 13 3.96 -3.96 7.92 0.015192975 30800 y 0.000646388

1.25 3 7 13 6.6 -6.6 13.2 0.032833181 55300 y 0.001396893

1.25 5 7 13 7.95 -7.95 15.9 0.04699375 71969.35716 y 0.001999357

1.25 1 9 13 3.96 -3.96 7.92 0.01251345 29400 y 0.000532387

1.25 3 9 13 6.6 -6.6 13.2 0.026580956 51200 y 0.001130891

1.25 5 9 13 7.95 -7.95 15.9 0.03855625 60133.31264 y 0.001640382

1.25 1 5 11 3.96 -3.96 7.92 0.015639563 34600 y 0.000665388

1.25 3 5 11 6.6 -6.6 13.2 0.032772356 73875 y 0.001394306

1.25 5 5 11 7.95 -7.95 15.9 0.047675 104771.8516 y 0.002028341

1.25 1 7 11 3.96 -3.96 7.92 0.015639563 37900 y 0.000665388

1.25 3 7 11 6.6 -6.6 13.2 0.035021161 64550 y 0.001489981

1.25 5 7 11 7.95 -7.95 15.9 0.05105 119359.4216 y 0.002171931

1.25 1 9 11 3.96 -3.96 7.92 0.015192975 34400 y 0.000646388

1.25 3 9 11 6.6 -6.6 13.2 0.033875009 62300 y 0.001441218

1.25 5 9 11 7.95 -7.95 15.9 0.01763125 83977.88474 y 0.000750125

1.25 1 5 9 3.96 -3.96 7.92 0.013853213 33700 y 0.000589387

1.25 3 5 9 6.6 -6.6 13.2 0.027994666 68200 y 0.001191038

50

1.25 5 5 9 7.95 -7.95 15.9 0.0106625 95892.75035 y 0.000453638

1.25 1 7 9 3.96 -3.96 7.92 0.013183331 35800 y 0.000560887

1.25 3 7 9 6.6 -6.6 13.2 0.028433194 69362.5 y 0.001209695

1.25 5 7 9 7.95 -7.95 15.9 0.01246875 131338.0119 y 0.000530485

1.25 1 9 9 3.96 -3.96 7.92 0.015192975 39600 y 0.000646388

1.25 3 9 9 6.6 -6.6 13.2 0.036623812 78437.5 y 0.001558167

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1.75 3 5 13 6.6 -6.6 13.2 0.086015043 99594.9375 y 0.003659525

1.75 5 5 13 7.95 -7.95 15.9 0.1207125 225927.225 y 0.005135735

1.75 1 7 13 3.96 -3.96 7.92 0.02503125 35850 n 0.001064959

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1.75 3 9 13 6.6 -6.6 13.2 0.065981908 71314.35625 y 0.002807212

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1.75 1 5 11 3.96 -3.96 7.92 0.027906745 34106.25 n 0.001187297

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1.75 1 7 11 3.96 -3.96 7.92 0.02358674 38986.5625 n 0.001003502

1.75 3 7 11 6.6 -6.6 13.2 0.079980275 113450 y 0.003402775

1.75 5 7 11 7.95 -7.95 15.9 0.11205 392795.5668 y 0.004767187

1.75 1 9 11 3.96 -3.96 7.92 0.02243125 38128.1875 n 0.000954341

1.75 3 9 11 6.6 -6.6 13.2 0.077301045 86100 y 0.003288787

1.75 5 9 11 7.95 -7.95 15.9 0.07813125 191024.8429 y 0.003324108

1.75 1 5 9 3.96 -3.96 7.92 0.02626875 34375.9375 n 0.001117609

1.75 3 5 9 6.6 -6.6 13.2 0.069617263 111500 y 0.002961879

51

1.75 5 5 9 7.95 -7.95 15.9 0.0644625 282429.9014 y 0.002742568

1.75 1 7 9 3.96 -3.96 7.92 0.021878472 35770.9375 n 0.000930823

1.75 3 7 9 6.6 -6.6 13.2 0.067129523 121512.5 y 0.002856037

1.75 5 7 9 7.95 -7.95 15.9 0.06561875 413651.6703 y 0.002791761

1.75 1 9 9 3.96 -3.96 7.92 0.021125 38150 n 0.000898767

1.75 3 9 9 6.6 -6.6 13.2 0.079524509 136187.5 y 0.003383384

1.75 5 9 9 7.95 -7.95 15.9 0.08005 972699.8381 y 0.003405741

1 2 5 13 5.44125 -5.44125 10.8825 0.0153 40300 y 0.000650941

1 4 5 13 7.43625 -7.43625 14.8725 0.0243 59800 y 0.001033848

1 2 7 13 5.44125 -5.44125 10.8825 0.0138 34313 y 0.000587123

1 4 7 13 7.43625 -7.43625 14.8725 0.0222 49763 y 0.000944503

1 2 9 13 5.44125 -5.44125 10.8825 0.0114 33763 y 0.000485015

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1 2 5 11 5.44125 -5.44125 10.8825 0.0136 41463 y 0.000578614

1 4 5 11 7.43625 -7.43625 14.8725 0.0212 61213 y 0.000901958

1 2 7 11 5.44125 -5.44125 10.8825 0.0152 44813 y 0.000646687

1 4 7 11 7.43625 -7.43625 14.8725 0.025 67163 y 0.001063629

1 2 9 11 5.44125 -5.44125 10.8825 0.0149 39575 y 0.000633923

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1 2 5 9 5.44125 -5.44125 10.8825 0.0118 41463 y 0.000502033

1 4 5 9 7.43625 -7.43625 14.8725 0.0184 60213 y 0.000782831

1 2 7 9 5.44125 -5.44125 10.8825 0.0124 44125 y 0.00052756

1 4 7 9 7.43625 -7.43625 14.8725 0.0198 65825 y 0.000842394

1 2 9 9 5.44125 -5.44125 10.8825 0.016 53050 y 0.000680723

1 4 9 9 7.43625 -7.43625 14.8725 0.0266 82250 y 0.001131702

2 2 5 13 5.44125 -5.44125 10.8825 0.082 51965 y 0.003488704

2 4 5 13 7.43625 -7.43625 14.8725 0.146 240021 y 0.006211596

52

2 2 7 13 5.44125 -5.44125 10.8825 0.0694 51590 y 0.002952635

2 4 7 13 7.43625 -7.43625 14.8725 0.1264 145546 y 0.00537771

2 2 9 13 5.44125 -5.44125 10.8825 0.0582 58463 y 0.002476129

2 4 9 13 7.43625 -7.43625 14.8725 0.1076 107613 y 0.004577861

2 2 5 11 5.44125 -5.44125 10.8825 0.078 44825 y 0.003318524

2 4 5 11 7.43625 -7.43625 14.8725 0.1358 337725 y 0.005777635

2 2 7 11 5.44125 -5.44125 10.8825 0.0729 28585 y 0.003101543

2 4 7 11 7.43625 -7.43625 14.8725 0.1345 415129 y 0.005722326

2 2 9 11 5.44125 -5.44125 10.8825 0.0678 45300 y 0.002884563

2 4 9 11 7.43625 -7.43625 14.8725 0.1304 191000 y 0.005547891

2 2 5 9 5.44125 -5.44125 10.8825 0.0702 46736 y 0.002986671

2 4 5 9 7.43625 -7.43625 14.8725 0.1184 311080 y 0.005037349

2 2 7 9 5.44125 -5.44125 10.8825 0.0639 41310 y 0.002718637

2 4 7 9 7.43625 -7.43625 14.8725 0.1163 392554 y 0.004948004

2 2 9 9 5.44125 -5.44125 10.8825 0.0681 81545.99048 y 0.002897327

2 4 9 9 7.43625 -7.43625 14.8725 0.1325 440789.3784 y 0.005637236

1.5 2 5 13 5.44125 -5.44125 10.8825 0.0446 62375 y 0.001897515

1.5 4 5 13 7.43625 -7.43625 14.8725 0.074 97375 y 0.003148343

1.5 2 7 13 5.44125 -5.44125 10.8825 0.0386 51125 y 0.001642244

1.5 4 7 13 7.43625 -7.43625 14.8725 0.0648 75825 y 0.002756927

1.5 2 9 13 5.44125 -5.44125 10.8825 0.0318 50700 y 0.001352937

1.5 4 9 13 7.43625 -7.43625 14.8725 0.0538 65700 y 0.00228893

1.5 2 5 11 5.44125 -5.44125 10.8825 0.0403 68625 y 0.001714571

1.5 4 5 11 7.43625 -7.43625 14.8725 0.0669 120525 y 0.002846272

1.5 2 7 11 5.44125 -5.44125 10.8825 0.0412 55763 y 0.001752861

1.5 4 7 11 7.43625 -7.43625 14.8725 0.0694 120713 y 0.002952635

1.5 2 9 11 5.44125 -5.44125 10.8825 0.0452 57650 y 0.001923042

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

1.5 4 5 9 7.43625 -7.43625 14.8725 0.0412 106862.5 y 0.001752861

1.5 2 7 9 5.44125 -5.44125 10.8825 0.0393 54050 y 0.001672025

1.5 4 7 9 7.43625 -7.43625 14.8725 0.0427 149950 y 0.001816679

1.5 2 9 9 3.96 -3.96 10.8825 0.0467 30525 y 0.00198686

1.5 4 9 9 6.6 -6.6 14.8725 0.0549 249825 y 0.00233573

1.25 2 5 13 3.96 -3.96 10.8825 0.0284 47223.8 y 0.001208283

1.25 4 5 13 6.6 -6.6 14.8725 0.0458 77228.2 y 0.001948569

1.25 2 7 13 3.96 -3.96 10.8825 0.0247 44028.68 y 0.001050866

1.25 4 7 13 6.6 -6.6 14.8725 0.0411 64612.72 y 0.001748607

1.25 2 9 13 3.96 -3.96 10.8825 0.0197 41907.8 y 0.00083814

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

1.25 4 5 11 6.6 -6.6 14.8725 0.0402 90372.2 y 0.001710316

1.25 2 7 11 3.96 -3.96 10.8825 0.0259 47705.18 y 0.00110192

1.25 4 7 11 6.6 -6.6 14.8725 0.0439 88434.56 y 0.001867733

1.25 2 9 11 3.96 -3.96 10.8825 0.0288 49127.96 y 0.001225301

1.25 4 9 11 6.6 -6.6 14.8725 0.0296 73916.84 y 0.001259337

1.25 2 5 9 3.96 -3.96 10.8825 0.025 51801.36 y 0.001063629

1.25 4 5 9 6.6 -6.6 14.8725 0.0238 82898.44 y 0.001012575

1.25 2 7 9 3.96 -3.96 10.8825 0.0247 49029.8 y 0.001050866

1.25 4 7 9 6.6 -6.6 14.8725 0.0243 96798.4 y 0.001033848

1.25 2 9 9 3.96 -3.96 10.8825 0.0309 47534 y 0.001314646

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

1.75 2 5 11 3.96 -3.96 10.8825 0.0567 63030 y 0.002412311

1.75 4 5 11 6.6 -6.6 14.8725 0.0971 204338 y 0.004131137

1.75 2 7 11 3.96 -3.96 10.8825 0.0551 50608 y 0.002344239

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

1.75 4 5 9 6.6 -6.6 14.8725 0.0724 185243.4 y 0.003080271

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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