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Developments toward a micro Bistable Aerial Platform

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Title:
Developments toward a micro Bistable Aerial Platform analysis of the Quadrantal Bistable Mechanism
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English
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Muñoz, Aaron A
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University of South Florida
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Subjects / Keywords:
Microelectromechanical systems (MEMS)
Compliant
Out-of-plane
Ortho-planar
3-D mechanism
Pop-up structure
Dissertations, Academic -- Mechanical Engineering -- Masters -- USF   ( lcsh )
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non-fiction   ( marcgt )

Notes

Abstract:
ABSTRACT: The Bistable Aerial Platform (BAP) has been developed in order to further enlarge the repertoire of devices available at the microscale. This novel device functions as a switch in that its platform can lock in two positions, up or down. Herein, it will be examined and explained, but a true understanding of its workings requires a better understanding of its compliant constituent parts. The Helico-Kinematic Platform (HKP), which serves as an actuator for the BAP, is currently under investigation by another researcher and will be merely touched upon here. The focus, therefore, will rest on the analysis of the Quadrantal Bistable Mechanism (QBM), the principle component of the BAP. A preliminary pseudo-rigid-body model, an aid for the understanding of compliant mechanisms, will also be examined for the QBM. The models developed for these two devices, the HKP and QBM, can later be combined to form a full model of the Bistable Aerial Platform.
Thesis:
Thesis (M.S.M.E.)--University of South Florida, 2008.
Bibliography:
Includes bibliographical references.
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Mode of access: World Wide Web.
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by Aaron A. Mũnoz.
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Title from PDF of title page.
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Document formatted into pages; contains 95 pages.

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ABSTRACT: The Bistable Aerial Platform (BAP) has been developed in order to further enlarge the repertoire of devices available at the microscale. This novel device functions as a switch in that its platform can lock in two positions, up or down. Herein, it will be examined and explained, but a true understanding of its workings requires a better understanding of its compliant constituent parts. The Helico-Kinematic Platform (HKP), which serves as an actuator for the BAP, is currently under investigation by another researcher and will be merely touched upon here. The focus, therefore, will rest on the analysis of the Quadrantal Bistable Mechanism (QBM), the principle component of the BAP. A preliminary pseudo-rigid-body model, an aid for the understanding of compliant mechanisms, will also be examined for the QBM. The models developed for these two devices, the HKP and QBM, can later be combined to form a full model of the Bistable Aerial Platform.
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DEVELOPMENTS TOWARD A MICRO BISTABLE AERIAL PLATFORM: ANALYSIS OF THE QUADRANTAL BISTABLE MECHANISM by Aaron A. Mu–oz A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Department of Mechanical Engineering College of Engineering University of South Florida Major Professor: Craig P. Lusk, Ph.D Autar K. Kaw, Ph.D Nathan B. Crane, Ph.D Date of Approval: October 30 2008 Keywords: microelectromechanical systems (MEMS), complia nt, out of plane, ortho planar, 3 D mechanism, pop up structure, pseudo rigid body model Copyright 2008, Aaron A. Mu–o z

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ACKNOWLEDGMENTS I would like to express my utmost appreciation to Dr. Craig Lusk who gave me the opportunity to join in his research and to benefit from his wisdom I would like to thank him for the guidance he has given me and for the many hours he spent on my behalf to ensure my success I will always be grateful. Also, I would like to thank my graduate committee members, Dr. Nathan Crane and Dr. Autar Kaw for thei r commitment to helping me improve this thesis. Furthermore, I would also like to thank my f ather who inspired me to be come an engineer and to the rest of my family for their support throughout my education. Additionally, I would like to thank the many people in the College of Engineering and the Mechanical Engineering Department at the University of South Florida that make this university great This work was supported, in part, by the University of South Florida Office of Research & Innovation through the New Researcher Grant Program under Grant Number R057524. Finally I am thankful to my Heavenly Father for His love, care, and guidance through all the stages of my l ife including this research.

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! i TABLE OF CONTENTS LIST OF FIGURES iii ABSTRACT v CHAPTER 1: INTRODUCTION 1 1.1. Objective 1 1.2. Motivation 2 1.3. Scope 3 1.4. Contributions 3 1.5. Research Approach 4 CHAPTER 2: BACKGROUND 5 2.1. MEMS 5 2.2. Compliant Mechanisms 6 2.3. Pseudo Rigid Body Model 6 2.3.1. Fixed Free Flexible Cantilever 7 2.3.2. Fixed Guided Flexible Beam 8 2.4. Bistability 10 2.5. Spherical Mechanisms 11 CHAPTER 3: BISTABLE AERIAL PLATFORM 12 3.1. Compliant Components 12 3.1.1. Quadrantal Bistable Mechanism 12 3.1.2. Compliant Helico Kinematic Platform 14 3.2. Inception of the Bistable Aerial Platform 16 3.2.1. QBM Pair 16 3.2.2. HKP Actuation 18 3.2.3. Platform Integration 21 CHAPTER 4: MICRO BI STABLE AERIAL PLATFORM 23 4.1. MEMS Prototype 23 4.2. Die Release 27 4.3. Testing Results 31 CHAPTER 5: QBM FORCE RELATIONSHIPS 34 5.1. Restrained ANSYS Model 34 5.2. Coordinate Transformation 36 5.3. Force Analysis 37 5.4. FEA of the Bistable Aerial Platform 40

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! ii CHAPTER 6: QBM PSEUDO RIGID BODY MODEL 42 6.1. Related PRBM 42 6.2. Preliminary PRBM of a Spherical Fixed Guided Beam 44 6.2.1. Unrestrained ANSYS Model 45 6.2.2. Adequacy of the PRBM 46 6.2.3. Possible PRBM Improvements 48 CHAPTER 7: RESULTS AND DISCUSSION 50 7.1. BAP Design 50 7.2. QBM PRBM 51 REFERENCES 52 APPENDICES 55 Appendix A: ANSYS Code for a Restrained QBM 56 Appendix B: ANSYS Code for an Unrestrained QBM 62 Appendix C: MATLAB Code for a Restrained QBM 68 Appendix D: MATLAB Code for an Unrestrained QBM 73 Appendix E: ANSYS Code for the BAP 79 Appendix F: Coordinate Transformat ion 89 F.1. Mathematics of Rotations 89 F.2. Rotation Matrices 91

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! iii LIST OF FIGURES Figure 1.1 Illustration of the Bistable Aerial Platform 2 Figure 2.1 (a) Cant ilever with a vertical force at the free end (b) Equivalent pseudo rigid body model 8 Figure 2.2 (a) Fixed guided flexible beam (b) Free body diagram of one half of the beam (c) Pseudo rigid body model 9 Figure 2.3 The "ball on the hill" analogy 10 Figure 3.1 Top view of Quadrantal Bistable Mechanism 13 Figure 3.2 Side view of Quadrantal Bistable Mechanism 13 Figure 3.3 Linear beam buckled under a compressive load 14 Figure 3.4 Top view of Helico Kinematic Platform 15 Figure 3.5 Side view of actuated HKP 15 Figure 3.6 Planar threshold force vs. 16 Figure 3.7 Conjoined QBM Pair 17 Figure 3.8 Integrated compliant components of the BAP 19 Figure 3.9 Elevating ring lifting the ortho planar links 20 Figure 3.10 Top view of complete BAP mechanism showing the attached platform 22 Figure 3.11 Side view of the BAP platform in its initial, intermed iate, & final positions 22 Figure 4.1 L Edit design file of a Bistable Aerial Platform 24 Figure 4.2 L Edit design file of the small BAP variation 25 Figure 4.3 Typical MEMS staple hinge 26 Figure 4.4 Micro BAP mechanism with guide ring 27 Figure 4.5 BAP seen through optical microscope 29 Figure 4.6 Small BAP seen through SEM 30 Figure 4.7 QBM seen through SEM 30 Figure 4.8 Partly actuated BAP with partially raised platform 31 Figure 4.9 Small BAP with broken handle 32 Figure 5.1 Flowchart of the restrained ANSYS model 35 Figure 5.2 Rotating frames on the quadrantal beam 36 Figure 5.3 Planar threshold force & HKP force response vs. 39 Figure 5.4 Ortho planar link & HKP leg raised by 40 Figure 6.1 (a) Compliant planar slider with straight slides (b) Compliant spherical slider with circular slides 43 Figure 6.2 PRBM for a compliant spherical slider 43 Figure 6.3 Angles me asured by the half model for the QBM 44 Figure 6.4 Full PRBM of the QBM 45 Figure 6.5 Ortho planar angle of the quadrantal beam's center vs. 47

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! iv Figure 6.6 Planar angle of the quadrantal beam's center vs. 47 Figure 6.7 Ratio of the moments to forces in the PRBM and the actual QBM 48 Figure 6.8 Planar angle vs. with linearly increasing arc length 49 Figure E.1 ANSYS Model of the BAP 79 Figure F.1 Rotation of a vector 90 Figure F.2 Rotating frames on the quadrantal beam 92

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! v ! DEVELOPMENTS TOWARD A MICRO BISTABLE AERIAL PLATFORM: ANALYSIS OF T HE QUADRANTAL BISTABLE MECHANISM Aaron A. Mu–oz ABSTRACT The Bistable Aerial Platform (BAP) has been developed i n order to further enlarge the repertoire of devices available at the microscale. This novel device functions as a switch in that its platform can lock in two positions, up or down. Herein, it will be examined and explained, but a true u nderstanding of its workings requires a better understanding of its compl i a nt constituent parts. The Helico Kinematic Platform (HKP) which serves as an actuator for the BAP, is currently under investigation by another researcher and will be merely touched upon here The focus therefore will rest on the analysis of the Quadrantal Bistable Mechanism (QBM) the principle component of the BAP. A preliminary pseudo rigid body model an aid for the understanding of compliant mechanisms will also be examined for the QBM The models developed for these two devices the HKP and QBM can later be combined to form a full model of the Bistable Aerial Platform

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! 1 CHAPTER 1: INTRODUCTION At the microscale, compliant mechanisms are of vital importance because frictional forces encountered in conventional rigid joints dominate [ the inertial forces ] at micro level thus making the use of rigid link mechanisms inappropriate for m icro applications [1] Because friction in the microscale discourages the use of gears and joints due to excessive energy loss, the obvious alternative choice is compliant mechanisms since they do not suffer frictional losses [2] A study of compliance is of particular importance to the further development of m icro e lectro m echanical s ystems (MEMS) because compliant mechanism s reduce part count s when compared with rigid body mechanisms that produce the same function thus enabling further miniaturization However, d ue to their use of large, nonlinear deflections their analysis proves far more demanding than an equivalent rigid body mechanism. For this reason, P seudo R igid B ody M odel s (PRBM s) are de veloped easing the analysis of new compliant mechanism s al lowing them to be modeled by an equivalent rigid body mechanism [3] 1.1. Objective The purpose of this research is to describe a novel compliant mechanism, the Bistable Aerial Platform (BAP), and to begin the develop ment of a pseudo rigid body model. Towards this end, a preliminary PRBM was examined for a critical component of the BAP, the Quadrantal Bistable M echanism (QBM). A finite element analysis (FEA) of b oth the QBM and BAP was also done (See A ppendi ces for ANSYS code ). A full PRBM of the BAP will be possible when complimentary research on the Helico Kinematic

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! 2 Platform (HKP), another key component that functions as an actuator for the BAP, is completed Figure 1 1 Illustration of the Bistable Aerial Platform 1.2. Motivation The Bistable Aerial Platform is a compliant mechanism that converts a rotational input into a large ortho planar displacement of a platform with two stable equilibrium positions (down and up). At the micro level, i t is unique in that it is the first MEMS platform that can maintain either its up or down position without a constant input force "#$ $%&'()*+ ,-.!$&/*

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! 3 due to bistability Additionally, the platform it self is among the largest surface micro machined surfaces to be raised out of plane. For these reasons, the device may ha ve applications in micro optics as mirrored platform in DLP projectors in tactile displays as a controlled array of Braille dot s or as a n integrated micro antenna that uses the raised platform ring for RF signals 1.3. Scope The B istable Aerial Platform is formed from the combination of two compliant mechanisms as well a n additional rigid body mechanism. Equations describing the motion s of the rigid body components of the BAP are easily derived, but the analysis required for the compliant components is much more involved. The pseudo rigid body model of the Helico Kinematic Platform [4] is currently under development by another researcher and thus lies outside the scope of this research This pap er will therefore focus on the Q uadrantal Bistable M echanism a device closely related to that studied by Le—n et al. [5] This research improves upon the existing PRBM which provides equations for beam deflection by incorporating elastic deformation of the beam into the model With the proper pseudo rigid body models for both the QBM and HKP the Bistable Aerial Platform will become easier to study and explain. 1.4. Contribution s The primary contribution of this research is the integration of the Quadrantal Bistable Mechanism and Helico Kinematic Platform to form the Bistable Aerial Platform based upon the recognition that the leg of a HKP can function as a dual output device. Subsequent contributions in volved detailed analyses of the QBM with th e aim of successfully prototyping the BAP. First a finite element analysis was performed on the

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! 4 BAP and two finite element analyses were done on the QBM. One FEA of the QBM was done to confirm that this application within the BAP correlated to that done b y Le—n et al. [5] ; the other FEA was used to gather data on the elastic deformation experienced by the QBM. To analyze the FEA data, MATLAB code was produced and used to improve the related pseudo rigid body model developed by Le—n et al. [5] 1.5. Research Approach All information presented up to this point has served to introduce the subject matter of this thesis. In the next chapter, the relevant background information needed to better understand later portions of this thesis will be presented. Following that, this paper will follow the actual research approach undertaken. Chapter T hree will describe the Bistable Aerial Platform and how it was developed. Then, in the next chapter, the MEMS prototype of this device will be examined. Following this the focus will shift to the QBM. Chapter F ive will explain a force relations hip for the QBM that is crucial to the operation of the BAP. Then in C hapter S ix, a preliminary PRBM of a spherical fixed guided beam which corresponds to the QBM, will be introduced and analyzed. Finally, in C hapter S even, the overall BAP design will be discussed and the r esults concern ing the PRBM will be summarized

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! 5 CHAPTER 2: BACKGROUND 2.1. MEMS The current push for miniaturization, as witnessed in electronics as the quest for the ever smaller microprocessor, has recently spawned a novel field of micro machines, better known as m icroelectro mechanical s y stems (MEMS), which are continually becoming more commonplace as additional technologies and applications continue to develop. Micromanufacturing, as a technology, is relatively new; it was begun in the mid 1960's [6 ] and did not branch off into the MEMS process until 1987 [2] In its few scant decades of existence, microelectromechanical systems development has come a long way. The fabrication approach of MEMS is one that conveys the advantages of miniaturization, multiplicity, and microelectronics to the design and construction of integrated electromechanical systems [2] [7] Miniaturization is of course an inherent aspect of MEMS as is implied by its name; and w ith batch processes such as surface micromachining now allowing many mechanisms to be made for the same cost as a single device, multiplicity can really come into play for MEMS. This is important because it enables an array of single devices to produce macroscale effects that would be impossible for a discrete device [8] Additionally the MEMS process makes use of the same fabrication techniques and materials used for microelectronics, which enables the integration of both mechanical and electrical components [7]

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! 6 2.2. Compliant Mechanisms Compliant mechanisms are devices that gain their mobility from elastic deformation rather than the rigid body motions of conventional mechanisms [9] [10] Unlike traditional rigid link mechanisms where elastic deformation is detrimental to performance, a c ompliant mechanism is designed to take advantage of the flexibility of the material [11] The function of the compliant member within a compliant mechanism can be as basic as serving as a simple spring or as complex as generating a specified motion [12] The fully compliant mechanism uses flexible structures to emulate t he overall performance of rigid link mechanisms with without any rigid joints whatsoever [13] Compliant mechanisms are well suited fo r MEMS applications because their joint less, single piece construction is unaffected by many of the difficulties associated with MEMS, such as wear, fr iction, inaccuracies due to backlash, noise, and clearance problems associated with the pin joints [13] In addition, compliant mechanisms are planar in nature, do not require assembly, and can be made using a single layer [13] This greatly enhances the manufacturability of micromechanisms because MEMS are planar and are typically built in batch productio n with minimal or no assembly [1] 2.3. Pseudo Rigid Body Model A simpl e, accurate method for modeling compliant mechanisms is th e pseudo rigid body model This model functions by replacing flexible members with conventional kinematic members; thus, allowing the designer to model the compliant mechanism using rigid body equat ions [3] Most, but not all, of the typical configurations of flexible segments can be analyzed using this method.

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! 7 When an element bends, its ends follow a certain path. Therefore, what the pseudo rigid body model does is attempt to accurately model the behavior of a compliant mechanism by replacing a flexible segment with rigid segments and pin joints at optimized loc ations that will allow the end to follow the same deflection path. The stiffness of the flexible segment is balanced by the addition of torsional spring s because, "unlike rigid body mechanisms, energy is not conserved between the input and output port of c ompliant mechanisms because of energy storage in the flexible members" [14] 2.3.1. F ixed F ree Flexible C antilever As an example of the p seudo rigid b ody m odel, the flexible cantilever beam will be examined as it is one of the mo re co mmon application s of the PRBM A flexible cantilever beam with a constant cross section is show n in Figure 2 1 (a) [15] Note that the application of a force orthogonal to the beam at the free end will cause the end of the beam to deflect and follow a nearly circular path. The PRBM assumes that this nearly circular path can be modeled b y two beams that are pinned together forming a pivot This characteristic pivot is positioned a fractional distance l from the free end of the beam where is the characteristic radius factor of the length, l Thus making l the characteristic radius that represents the length of the pseudo rigid body link and radius of the circular deflection path it travels [15] In order to incorporate the beam's resistance to deflection into the model, a torsional spring is added at the cha racteristic pivot. T he resulting pseudo rigid body model shown in Figure 2 1 (b), will have approximately the same deflection path and vertical force response as the original compliant structure.

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! 8 Figure 2 1 (a) Cantilever with a vertical force at the free end (b) Equivalent pseudo rigid body model [ 5 ] 1 2.3.2. F ixed G uided F lexible B eam The PRBM for the fixed guided flexible beam is closely related to the fixed free flexible cantilever. The difference is that the angle of the "guided" end does not change, which requires a moment, M o be present at the guided end [15] This causes the deflected shape of the beam t o be anti symmetrical at its centerline as shown in Figure 2 2 (a). Note that the midpoint has zero curvature, which implies that the moment at the midpoint of the beam is also zero because moments are directly proportional to the cu rvature, per the Bernoulli Euler equation [15] Without a moment at the midpoint, the free body diagram for one half of the beam is as shown in Figure 2 2 (b), which matches the flexible cantilever. Thus the PRBM for half of the fixed guided beam will be the same as the PRBM for the fixed free beam. Therefore, combining two anti symmetric one half beams, as shown in Figure 2 2 (c), will form the PRBM for the entire beam and ensure that no moment is created at the center [15] !!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 1 Image used with permission.

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! 9 Figure 2 2 (a) Fixed guided flexible beam (b) F ree body diagram of one half of the beam (c) P seudo rigid body model [ 1 5 ] 2 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 2 Image adapted from Howell

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! 10 2.4. Bistability A system is considered to be stable if a small external disturbance only causes it to oscillate about its current position. The position is unstable if that same disturbance ca uses the system to move to another position [15] An example of these states is shown below in Figure 2 3 Figure 2 3 The "ball on the hill" analogy Positions A and C are stable positions. Position B is unstable. [ 1 5 ] 3 Typical mechanisms are monostable, having only one stable state, and require a sustained force in order to hold a second stable state [16] A bistable mechanism on the other hand, is capable of holding one of two stable states at any given time, and consumes energy only during the motion from one stable state to the other [16] This bistable behavior is achieved "by storing energy during part of its motion, and then releasing it as the mechanism moves toward a second stable state" [17] B ecause flexible segments store !!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 3 Image adapted from Howell

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! 11 energy as they deflect, compliant mechanisms can be designed to use the same segments to gain both motion and a second stable state, which results in a significant reduction in part count [17] 2.5. Spherical Mechanisms In planar mechanisms, the joint axes are parallel to each other and norma l to the p lane in which the mechanism undergoes its motion I n spherical mechanisms, however, the joint axes all intersect at the center of a sphere which caus es each link to rotate about this fixed point in space [4] [18] A spherical mechanism therefore require s that the axes of all joint s fixed in space to the ground (or reference) link intersect at a common point Additionally, mobile joint axes not connected to ground which change their orientation in space must be constrained by the ir linkage geometry to pass through that same center point [4] The Quadrantal Bistable Mechanism and the Helico Kinematic Platform, which form the Bistable Aerial Platform, are both spherical mechanisms. Therefore, the BAP itself is a partial spherical mechanism in which most of the joint axes rotate about the mechan ism's center.

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! 12 CHAPTER 3: BISTABLE AERIAL PLATFORM 3.1. Compliant Components In order to understand the BAP, it is necessary to describe its mechanical components. This mechanism is comprised of three building block elements. The first element is a Quadrantal Bistable Mech anism (QBM), which is the cornerstone of the BAP and closely relates to a simplified spherical PRBM developed at USF [19] [20] The second is a compliant Helico Kinematic Platform [4] that serves to actuate the QBM. T he final element is a variation of the scissor lift mechanism that attaches to the output of the QBM and amplifies the ortho planar displacement. 3.1.1. Quadrantal Bistable Mechanism The Quadrantal Bistable Mechanism consists of two links that rotate about intersecting, orthogonal axes. The rotation of one link (the planar link ) is in plane while the other (the ortho planar link ) rotates out of plane. The axes of rotation of the planar and ortho planar links are called the planar axis and ortho planar ax i s respectively. In the QBM's initial position, the links are perpendicular to each other and lie in (or near) a common base plane. A thin, compliant beam, the quadrantal beam connects the two links and forms an arc that is approximately a quarter circle (or quadrant) See Figure 3 1 and Figure 3 2

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! 13 Figure 3 1 Top view of Quadrantal Bistable Mechanism Figure 3 2 Side view of Quadrantal Bistable Mechanism There are two potential input links for the QBM. Either the planar or the ortho planar (OP) link may be used either individually or simultaneously. Regardless o f the mode of actuation, the mechanism can move from its initial (first stable) position to its second stable position, which occurs when the ortho planar link reaches ninety degrees of rotation.

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! 14 Equations defining the QBM's position have already been dev eloped for the case where the dr iving link rotates out of plane [20] but work has yet to be done for the case of the planar driving link. This is partly because a relatively large force, the planar threshold force is required to actuate the mechanism using the planar link. The benefit of actuating the device in this manner, however, is that with only a few degrees of rotation, the ortho planar link is drawn through a full ninety degrees into its second stable posi tion. Fortunately, providing a bias to the mechanism by first raising the ortho planar link greatly reduces the planar threshold force The higher the ortho planar link is raised, the less force is needed to rotate the planar link ; this force phenomenon wi ll be further examined in chapter five 3.1.2. C ompl i a nt Helico Kinematic Platform The compliant Helico Kinematic Platform (HKP) is a spherical mechanism in which a platform is raised out of plane by the coordinated buckling of beams. Shown in Figure 3 3 is a long, thin beam fixed at one end and with the motion of its other end limited by a longitudinal slide Upon ap pl y ing a longitudinal, compressive force to the beam at the slide the beam will buckle and displace its center. By t aking two or more of these beams and attaching a common platform at their centers their displacements will be constrained such that the beam s will only buckle out of plane thus raising the platform Curving these beams around a circular platform will allow these them to be Figure 3 3 Linear beam buckled under a compressive load

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! 15 simultaneously ac tuated by a single rotary input ( See Figure 3 4 ). When buckled, these beams, legs support the HKP like the legs of a table ( See Figure 3 5 ) Figure 3 4 Top view of Helico Kinematic Platform Figure 3 5 Side view of actuated HKP

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! 16 3.2. Inception of the B istable A erial P latform The Bistable Aerial Platform result ed from probative examinations of the Quadrantal Bistable Mechanism. Of particular note was the behavioral response usin g the planar link as the input A d e creasing planar threshold force was obtained by biasing the ortho planar link with an initial ortho planar displacement (See Figure 3 6 ) The o bservation of this phenomenon later led to the inspi ration to use of the HKP as a means of providing input simultaneously to both the planar and the ortho planar links. Figure 3 6 Planar threshold force vs 3.2.1. QBM Pair The continued investigation into the QBM was done with the objective of developing a new mechanism utilizing the QBM to provide bistability to some platform

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! 17 type mechanism. The initial thought was to use multiple QBMs positioned around the platform and hav e the many ortho planar links raise the platform. While this would make for a very stable platform, the concept was abandoned due to the challenges involved in simultaneously actuating separate QBMs. The chosen alternative was to make use of the spherical design of the QBM and merge two QBMs into a unified mechanism that share the same base plane, planar axis, ortho planar axis, and center point. To achieve this, o ne QBM was rotated by one hundred eighty degrees with respect to the other about their common planar axis ( See Figure 3 7 ) With the planar links further conjoined, only a single rotary input will be needed to si multaneously actuate both QBMs. Figure 3 7 Conjoined QBM Pair $%&0&*!1/02! 34)05)/0678 9*':) ; $%&0&* 1/02<

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! 18 3.2.2. HK P Actuation Actuation is accomplished through the use of a compliant Helico Kinematic Platform. Recall that the legs of the compliant HKP must b e compressed in order to buckle and that the simplest way to accomplish this is for one end to be fixed while a compressive force is applied to the other end. Also recall that the QBM, when actuated by rotating the planar link, without an out of plane bias of the ortho planar link, requ ires a very large planar threshold force to initiate movement. Therefore, the planar link is virtually fixed until the planar threshold force is reached, thereby allowing the attachme nt of the "fixed" end of a HKP leg to the planar link. In the BAP, the H KP acts as a transmission to provide input forces to the two QBMs The beauty of using this mechanism for actuation is that it can provide the planar threshold force and raise the ortho planar links to give the needed bias. This is accomplished by situatin g the HK P a spherical mechanism requiring rotary input, concentric with the QBMs Its platf orm is cut to form an annulus and designated as the elevating ring From this location ( S ee Figure 3 8 ) the elevating ring can concurrentl y raise the ortho planar links of both QBMs in order to re duce the planar threshold force Thus, w hen the HKP is rotated, its compliant legs are compressed by the "fixed" planar links and therefore buckle This forc es the elevating ring to rise and l ift the QBMs ortho planar links (See Figure 3 9 ) The elevating ring and ortho planar links will continue to rise until the planar threshold force decreases sufficiently to equal the force required to buckle the legs When this occurs the QBMs' planar links rotate causing the ortho planar links to be drawn upright and the elevating ring to suddenly collapse as its supports un

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! 19 buckle due to the withdrawal of the compressive force needed to maintain it in the raised position. These forc es are explained further in Chapter Five. Figure 3 8 Integrated compliant components of the BAP

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! 20 Figure 3 9 Elevating r ing lifting the ortho planar links When the QBMs links are upright, the QBM s are in their second stable position. Thus, the BAP is in its second stable position requiring no additional forces to maintain itself. Because this position is stable, the QBM's ortho planar links will resist forces smaller than the ortho planar threshold force Similar to the case for the planar threshold force, the magnitude of the ortho planar threshold force will depend on the position of the QBM's planar link. The compliant HKP again proves ideal b ecause it can also de actuate the BAP A simple reversal in the direction of the input forces on the HKP will put the legs under tension and pull the attached planar links of the QBMs. Once the planar links reach their ini tial position s the ortho planar links simply fall back down. Throughout this deactivation, the elevating ring will remain down. 9*':) ; $%&0&*!1/02< =%6>&'/0? @/0?

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! 21 3.2.3. Platform Integration The integral bistable mechanism of the Bistable Aerial Platform and its method of actuation have been described ; all that remains is to attach the actual platform. Note, t he elevating ring of the compliant HK P is not bistable nor does it lock in the up position The bistable platform of the BAP is a separate, additional platform that is connected to the ends of the two ortho planar links, as they are the only links that lock in the second stable position. As the BAP actuates, the ortho planar links of the QBM pair form a V shape whose interior angle becomes increasingly more acute This provides the requ ired input motion for a scissor lift mechanism. Connecting a traversing link to the ends of the ortho planar links and pinning them at their centers would form the elongating crossing pattern of a scissor lift as the device was actuated. However, in the fu lly actuated position this would provide merely a taller spike rather than the needed mount for a platform. Therefore rather than pin the new links in the center in the traditional fashion of a scissor lift the links will instead be joined at their ends by means of a large ring to serve as a platform (See Figure 3 10 ) Connected in this manner, the new links now have a shifting "pivot" point that allows them to conform to a platform rather than coming together at the center ( S ee Figure 3 11 ) The result is similar to that of the sliders used at the end of a scissor lift in typical aerial platforms.

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! 22 Figure 3 10 Top view of complete BAP mechanism showing the attached platform Figure 3 11 Side view of the BAP platform in its initial, intermediate, & final positions Initial In termediate Final

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! 23 CHAPTER 4: MICRO BISTABLE AERIAL PLATFORM 4.1. MEMS Prototype The B istable Aerial Platform was developed for possible micro applications in micro optics, tactile displays, or as a micro antenna This two position device, if built at the macroscale, would prove overly complex and impractical compared to an equivalent mechanism. At the microscale, however th is device is ideal because it is essentially flat and achieves its two positions with few mechanical parts thus reducing friction. The BAP prototype was built using the Multi User MEMS Process (MUMPs) one of the more common surface micromachining methods used in the microfabrication of MEMS, which introduce s the following constraints to th e design [16] [21] : 1. a maximum of 3 polysilicon layers built on top of a silicon substrate 2. a limited ability to create a design that involves sliding the topmost layer of polysilicon over the lower layer, due to the natural conformation of the top layer over the lower polysilicon layer during the fabrication proces s 3. an inability to create a design that involves structures with large moving surface areas (unanchored to the ground substrate layer), due to the likelihood that the structure will be carried away in the fabrication process 4. a minimum surface feature of 2 m 5. a minimum surface feature spacing of 2 m

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! 24 Shown in Figure 4 1 and Figure 4 2 are the design files for two variations of the BAP that conform to these constraints units are in micrometers ( m) Figure 4 1 L Edit design file of a Bistable Aer ial Platform

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! 25 Figure 4 2 L Edit design file of the small BAP variation With just three structural layers, only two of which are capable of making movable components, portions of the device had to be move d radially outwards because they could not be accommodated in their ideal location Such was the case for the planar links of the QBMs that must rotate together but can not be connected and pinned at the center due to interference from the hinges for the ort ho planar links Because the staple hinges used for the ortho planar links require both moveable layers (See Figure 4 3 : the

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! 26 first movable layer is held down by the second) pinning the planar links could not also be done at the sam e location. T his issue was resolve d by limiting the planar links' motion to planar rotation by means of a gui ded, external ring (See Figure 4 4 ) Figure 4 3 Typical MEMS staple hinge

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! 27 Figure 4 4 Micro BAP mechanism with guide ring 4.2. Die Release Both variations of the BAP d evice were fabricated by MEMSCAP using their PolyMUMPS process. The completed dies arrived covered with a final protective layer of photoresist and with the various silicon dioxide, sacrificial layers produced during the manufacturing still in place. These sacrificial layers secure all the loose components during transit and protect the more fragile components. Completed MEMS devices must therefore be "released" by immersing them in a bath of hydrofluoric acid in order to

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! 28 dissolve the sacrificial layers and free the st ructural members This is followed by several minutes in deionized water and the n alcohol in order to clear away residual acid These steps must be done with precision to prevent pieces of the device being washed away. The die is then subsequently heated i n an oven to reduce stiction The term stiction refers to the adhesion of the microstructure to adjacent surfaces [22] Typically, it occurs after release as "freed" members become stuck to the newly exposed surface underneath the sacri ficial layer [22] [23] Stiction becomes a vital issue in MEMS devices that incorporate large area, very thin, compliant members with a small offset from adjacent surfaces [22] [23] If the strength of the adhesive bonding resulting from this contact exceeds the vertical pull off force that can be generated by the structures, it will remain permanently stuck and be virtually anchored to the substrate [22] [24] The approach used to reduce or prevent stiction wa s to minimize the real area of contact through the formation of microscale standoff bumps ("dim ples") on one surface, which increases the nominal separation between the surfaces [24] [25] Shown below in Figure 4 5 is a released Bistable Aerial Platform viewed through a n optical microscope. In Figure 4 6 the smaller variation of the BAP is shown as observed through a scanning electron microscope (SEM). A released QBM, also viewed by a SEM is s hown in Figure 4 7

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! 29 Figure 4 5 BAP seen through optical microscope

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! 30 Figure 4 6 Small BAP seen through SEM Figure 4 7 QBM seen through SEM

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! 31 4.3. Testing Results This version of the BAP mechanism was designed to be manually actuated through the use of probes manipulators that allow the precise positioning of thin needles on the surface of a semiconductor device. A vacuum i s utilized to secure the die against the movements of the probes. The probe tips are position ed above the die and are then touched down on the die surface; thus allowing the released MEMS to be manipulated by subtle movements of the probe, which can push, pull, and prod the mechanism. Manipulating the mechanisms proved cumbersome, with the result being numerous broken devices. This is partly due to the fact that polysilicon shares many of the characteristi cs of glass and will tend to shatter. Fortunately, the effort was rewarded with a small measure of success. Show n below in Figure 4 8 is the larger variety BAP with a partial ly raised main platform. Unfortunately, further actuat ion of the mechanism Figure 4 8 Part ly actuated BAP with partially raised platform Note the gradational shading on the out of plane beams and that the platform has moved out of focus. Probe tip Beam moving out of plane Beam moving out of plane Platform

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! 32 immediately resulted in the fracture of the mechanism's handle Figure 4 9 shows the small BAP following actuation Figure 4 9 Small BAP with broken handle The arrow indicates the point of fracture. In both cases, the h andle sheared off at its thinnest section where it passes over the guide ring for the planar links. Additional probing of the broken mechanism indicates that stiction was not interfering with its operation. In all probability, the fracture came about due to excessive frict ion and a concentration of stress es where the polysilicon

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! 33 layers overlapped The d imples used to eliminate stiction also serve d to reduce sliding friction along the substrate (underlying surface). However, when using the handle to rotate the ring that serv ices the legs of the HKP, the edge of the ring is pressed up against its guides on the side opposite of the handle The resulting imbalanced distribution of friction al forces is believed to have eventually caused the ring to bind and the handle to then she ar off. In future iterations manually actuated micro BAPs should include a second handle directly opposite of the first one in order to offset the frictional imbalance, which causes binding. A better alternative would be to incorporate rotary thermal actu ators which would evenly distribute the force s keep the ring centered to avoid friction along the edges and remove the human factor.

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! 34 CHAPTER 5: QBM FORCE RELATIONSH I PS The Quadrantal Bistable Mechanism is the crucial component of the B istable A erial P latform. The basic concept for the BAP design evolved from an observed force relationship in the QBM Physical t esting of a QBM model shows that a n increase in the initial angle of the ortho plan a r link yield s a decrease in the re quisite planar threshold force. T herefo re, in order t o quantify this relatio n, a simplified ANSYS model was developed. 5.1. Restrained ANSYS Model The simplified model ( whose ANSYS code is given in Appendix A ) consists of only the quadrantal beam of the QBM which is the only point of interest for observing the force relationship between the initial bias angle of the ortho planar link and the planar threshold force required to actuate the QBM Within the model, the ortho planar end of the quadrantal beam was first displaced by a rotation the ang le of the ortho planar link relative to its initial position During the rotation of the ortho planar end, the position of the planar end was held constant. Following this ortho planar displacement, the ortho planar end 's new position was held constant whi le the planar end was displaced by a rotation of the planar link that increased from zero to five degrees of rotation in half degree increments which was sufficient to ensure that the plan ar threshold force had been met for all cases Once the planar end completed its motion, the beam reset to its initial posit ion and the subsequent case bega n with a new which increased incrementally by five degrees from zero to ninety ( See flow chart in Fi gure 5 1 ).

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! 35 Fi gure 5 1 Flowchart of the restrained ANSYS model The model was setup in this fashion in order to find the force applied to the QBM's planar link that will result in a reversal of the direction of the force on the ortho planar link. At this instant, the planar threshold force needed for actuation has been met and the QBM will tend towards its second stable position. The initial angle of the ortho planar link was varied so that the relation ship betw een and the planar threshold force

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! 36 could be observed. This is important because the BAP design requires an inverse relationship in order for the HKP to function as an actuator. Displacement loads were used in this model to insure that sufficient force ha d been applied to the planar links, since the planar threshold force s for the various bias angles were unknown prior to this analysis. 5.2. Coordinate Transformation The ANSYS data provides the forces experienced by the model in terms of the original xyz coordi nate system. However, in order to properly monitor the direction of the forces at the ends of the rotating links, t he data taken from ANSYS must be transformed from the original frame of reference, O, into new frames, A and B, that rotate with the planar a nd ortho planar links, respectively. The method used to accomplish t he transformation is further explained in Appendix F. The original and moving frames are shown below in Figure 5 2 Figure 5 2 Rotating frames on the quadrantal beam

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! 37 and will be transformed by ( 5 1 ) ( 5 2 ) T h is means that the A frame for the planar link represents a rotation of the O frame about the original z a xis by and that the B frame represents a rotation abo ut the x axis by Therefore, ( 5 3 ) and ( 5 4 ) These rotation matrices will allow the force data provided by ANSYS to be adjus ted for the planar and orth planar links, t hus allowing F x F y and F z to be seen in terms of the relative position and orientation of the link. 5.3. Force Analysis The observed decrease in the planar threshold force with an increased rotation of the ortho planar link by is fundamental to the operation of the BAP. To mathematically confirm this observation, the ANSYS data for the restrained model has been analyzed using MATLAB. In this analysis ( See Appendix C for MATLAB code ), the forces at the

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! 38 ends of the planar and ort ho planar links have been transformed into their relative frames using the rotation matrices in equations ( 5 3 ) and ( 5 4 ) The forces on the ortho planar link are of particular concern in confirming the QBM f orce relationship When the reaction force normal to the ortho planar link, F z,B (F z in the B frame), changes from downward to upward that is an indication that the planar threshold force on the planar link has been met and that the ortho planar link will be pulled upward by the continued movement of the planar link. The MATLAB code was therefore programmed to find the value of the planar force (the planar threshold force) at which the F z,B changes sign. This was done for ortho planar link rotations of 0 # # 90 in order to represent a wide range of initial biases applied to the ortho planar link. The results of this analysis are illustrated by the graph in Figure 5 3 which confirm s the QBM force relationship required for the BAP to function

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! 39 Figure 5 3 Planar threshold force & HKP force response vs. Also shown in Figure 5 3 is an assumed force response of a HKP leg within the BAP, raised by the same as shown in Figure 5 4 Another researcher is currently developing the PRBM of the HKP that will predict its actual force response but f or now howeve r, it is safe to assume that the force required to compress the HKP will increase with At some point, the force curves will intersect when the force required to further compress the HKP leg is equal to the decreasing planar threshold force of the QBM A t this shift point (the unstable position as described in section 2.4 ) the forces on the ortho planar links will tend to snap towards the ir second stable state, drawing the BAP's platform into the "up" position.

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! 40 Figure 5 4 O rtho planar link & HKP leg raised by 5.4. FEA of the Bistable Aerial Platform A finite element analysis was also done on the Bistable Aerial Platform i n order to confirm its general functionality (See Appendix E). The model used in this analysis (See Figure E 1 ) consisted of only the compliant components, the QBMs and the HKP. Contact elements were also included in order to permit the ortho planar links to be raised by the elevating ring in the manner previously described. As a displacement load compressed the l egs of the HKP, the elevating ring rose as expected and lifted the ortho planar links. However, in this model, which was based on the dimensions of the larger MEMS BAP, the planar force applied by the compressed Ortho Planar L ink HKP Leg Undeflected Position

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! 41 HKP legs never equaled the planar threshold force. Thus, it never achieved its second stable state. However, applying an additional displacement load directly to the planar links, following the partial actuation achieved with just the elevating ring, provided that requisite planar threshold force an d allowed the BAP to snap into its second stable state. This validates the basic concepts behind the design, but indicates that the QBMs and the HKP must be modified in order to properly balance the operant forces. Thus, a pseudo rigid body model is needed in order to better understand these compliant mechanisms.

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! 42 CHAPTER 6: QBM PSEUDO RIGID BODY M ODEL Pseudo rigid body models (PRBMs) are developed by observing the deflection path of a compliant mechanism and then determining an equivalent rigid body model that will follow the same approximate path. The chosen PRBM is then optimized until the error in the forces and deflection path between the compliant mechanism and its PRBM are minimized. In complex mechanisms, multiple models may be required before the error is bro ught to within acceptable tolerances. Therefore, i n an effort to accelerate the development of a PRBM for the QBM, a model that has already been developed for a similar mechanism [5] will be modified to match the QBM, thus reducing some of the initial design work 6.1. Related PRB M The related PRBM that will be used to form the new PRBM of the QBM was done for a similar spherical mechanism that consists of the same planar and ortho planar links with their corresponding axes, as well as a n arc connecting the two links. This particular model was designed as a compliant spherical slider ( S ee Figure 6 1 ) and addressed differing arc lengths and aspect ratios [5] It is a simplified case that assume s that the arc does not sag inward due to tensile forces during manipulation of the adjoining links. This is achieved by imposing an unrealistic constraint on the center of the arc that maintain s it at a constant radi al distance from the origin by means of an applied force. This prior model also differs from the QBM as the arc is pinned to the ortho planar link and not simply a single piece (See Figure 6 2 for PRBM).

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! 43 Figure 6 1 (a) Compliant p lanar s lider with straight slides (b) Compliant s pherical s lider with circular slides [5] 4 Figure 6 2 PRBM for a compliant spherical slider [5] 5 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 4 Image used with permission. 5 Image used with permission.

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! 44 6.2. Preliminary PRBM of a Spherical Fixed Guided Beam The result of pinning the deflected end in the fixed pinned PRBM model for the compliant spherical slider is the elimination of moments at the tip, which essentially causes it to behave as a free end with an applied force. Thus, the model developed by Le—n [20] is in actuality a spherical fixed free cantilever similar to the planar fixed free cantilever discussed in section 2.3.1 Furthermore, the QBM is essentially a spherical version of the fixed guided beam, also discussed in section 2.3.2 which implies that a similar relationship will exist between the compl iant spherical slider and the QBM as does between the planar fixed free and fixed guided beams. Therefore, the PRBM for the compliant spherical slider shown in Figure 6 2 will be used as a half model of the QBM to form the full mode l shown in Figure 6 4 This half model, with an arc length of forty five degrees, will predict the deflected position of the quadrantal beam's center (See Figure 6 3 ). Figure 6 3 Angles measured by the half model for the QBM

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! 45 Figure 6 4 Full PRBM of the QBM 6.2.1. Unrestrained ANSYS Model The intention of this ANSYS model for the QBM ( See Appendix B ) was to compare only the QBM's bending response to that of the preliminary PRBM of a spherical fixed free cantilever which currently does n o t address any type of elongation. Therefore, no displacement constraints were applied to the planar link so that the unres trained planar link would be free to rotate to its minimum energy position The model consists of the quadrantal beam and the planar link which limits one end of the

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! 46 quadrantal beam to in plane rotation about the z axis. The ortho planar end of the quadran tal beam is then raised out of plane by an angle which varies from 0 90. 6.2.2. Ade quacy of the PRBM The position analy s is of the ANSYS data for the small BAP and the proposed PRBM for a spherical fixed guided beam ( See Appendix D for MATLAB code ) was done with the results shown below. In Figure 6 5 the ortho planar angle, relative to the origin, of the quadrantal beam's center, is shown. Note that the last two sets of data perfectly overlap, which would be an ideal case where the s pherical quadrantal beam behaves the same as its planar cousin. The first set representing actual position data taken from the FEA, however, indicates that a significant amount of sag is occurring due to tensile forces. These tensile forces, which ac t along the quadrantal beam, cause it to flatten out its arc in order to accommodate the increased distance between the planar and ortho planar links, thus sagging inward. In Figure 6 6 this inward drift of the center point become s more apparent for the planar angle In spite of the problems caused by tensile force s which cause sagging, the deflections anticipated by the new, preliminary PRBM appear reasonable, especially since the points of interest will typically be the rigid lin ks which will not deform The forces and moments anticipated by the PRBM, h owever do not appear reasonable as is shown in Figure 6 7 This is caused not only by the deformation of the quadrantal beam due to tensile forces, which the prior PRBM neglected, but also by the prior PRBM's imposed constraint on the arc itself that maintained it at a constant radius. Therefore, the values of t he torsional spring constants must be adjusted to reflect these oversights.

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! 47 Figure 6 5 Ortho planar angle of the quadrantal beam's center vs. ! Figure 6 6 Planar angle of the quadrantal beam's center vs.

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! 48 Figure 6 7 Ratio of the moments to forces in the PRBM and the actual QBM 6.2.3. Possible PRBM Improvements Modifications to the PRBM must be made in order to more accurately model the elongating quadrantal beam of the QBM. One possible alteration is to have the overall length of the model increase as increases. A quick attempt at this was tried with the results shown in Figure 6 8 As indicated by the graph, linearly increasing the arc length, $ with increasing causes the planar angle to more closely match the actual planar angle of the compliant quadrantal beam. However, the relation is most li kely non linear, and further research will be required to discover t he proper relationship.

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! 49 Figure 6 8 Planar angle vs. with linearly increasing arc length Another possible way of accounting for the increasing arc length would be through the addition of another spring at the center of the arc to allow the beam to stretch. This spring approach, while more complex, may prove the better method because it can also address the condition when a force is applied on the planar link of the QBM, such as in the BAP. Accurately modeling the forces and moments is probably even more important than c ap turing the position of the beam Therefore, adjusting the spring cons tants for the torsional sprin gs and this additional spring meant to capture the energy stored by the elongation of the quadrantal beam should be the primary focus in completing the PRBM. !=45(1+"/100)

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! 50 CHAPTER 7: RESULTS AND DISCUSSION 7.1. BAP Design While the current iteration of the BAP will require additional refinement to reach its second state, the principle behind the design remains valid. The QBM force relationship, the fundamental theory on which the QBM is based, has been proven. Additionally, both the MEMS prototype and the finite element analysis of th e BAP itself (See Appendix E ) show the desired upward movement of the platform as the HKP lifts the ortho planar links. Unfortunately, the handle breaks during actuation of the prototype and the FEA model shows that the forces applied by the HKP legs on the planar links are insufficient to provide the necessary planar threshold force. Once the HKP and QBM forces are balanced, the BAP should move smoothly into its second state. However, because the compliant mechanisms involved are not yet fully understood and their PRBMs are not yet fully developed, properly balancing the forces to improve the mechanism can only be achieved through the trial and error of adjusting the arc length, aspect ratio, radius, etc. In the meant ime, a n additional rotational displacement load can be applied directly to the planar links once the BAP is partially actuated, to provide the necessary planar threshold force The ANSYS model for the large MEMS BAP prototype currently indicates that a sm all rotary displacement of the planar link is all that is required to get the mechanism to its "up" stable position. This implies that only a few subtle changes to the BAP are required to ach ieve the desired result unaided The FEA model of the BAP therefo re further proves the validity of the

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! 51 overall design and also serves as a justification for further research into compliant mechanisms 7.2. QBM PRBM Great progress has been made toward the development of a PRBM for the QBM A preliminary PRBM of a spherical fi xed guided beam has been developed that in its current state reasonably models the positional bending response of the QBM. However, further development is needed to capture the force and moment response and to model the elongation of the quadrantal beam due to tensile forces. Once the PRBM is perfected it will serve to balance the forces within the BAP by adjusting various physical aspects of the QBM and will work in conjunction with the PRBM of the HKP, which is also currently under development.

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! 52 REFERENCES [1] Ananthasuresh, G. K., Sridhar Kota, and Yogesh Gianchandani. "Systematic Synthesis of Microcompliant Mechanisms Preliminary Results." University of Michigan, 1993. [2] Fujita, Hiroyuki. "A Decade of MEMS and its Future." In stitute of Industrial Science, The University of Tokyo, 1997. [3] Howell, Larry L. "A Generalized Loop Closure Theory for the Analysis and Synthesis of Compliant Mechanisms." PhD Thesis, Purdue University, West Lafayette, 1993. [4] Lusk, Craig P. "Ortho Planar Mechanisms for Microelectromechanical Systems." PhD Thesis, Brigham Young University, Provo, 2005. [5] Le—n, Alejandro, Saurabh Jagirdar, and Craig P. Lusk. "A Pseudo Rigid Body Model for Spherical Mechanisms: The Kinematics of a Compliant Curved Beam." Proceedings of IDETC/CIE. New York: ASME, 2008. [6] Frazier, Bruno, Robert O. Warrington, and Craig Frie drich. "The Miniaturization Technologies: Past, Present, and Future." IEEE Transactions on Industrial Electronics 42, no. 5 (1995): 423 430. [7] Gabriel, Kaigham J. "Microelectromechanical Systems (MEMS) Tutorial." (Carnegie Mellon University) 1998. [8] Koester, David A., Karen W. Markus, and Mark D. Walters. "MEMS: Small machines for the microelectronics age." (MCNC MEMS Technology Applications Center) January 1996: 93 94. [9] Ananthasuresh, G. K., and Sridhar Kota. "Designing Compliant Mechanisms." Mec hanical Engineering Nov. 1995: 93 96. [10] Howell, L. L., A. Midha, and T. W. Norton. "Evaluation of Equivalent Spring Stiffness for Use in a Pseudo Rigid Body Model of Large Deflection Compliant Mechanisms." Journal of Mechanical Design 118 (March 1996) : 126 131.

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! 53 [11] Canfield, S. L., J. W. Beard, N. Lobontiu, E. O'Malley, M. Samuelson, and J. Paine. "Development of a Spatial Compliant Manipulator." International Journal of Robotics and Automation 17, no. 1 (2002): 63 70. [12] Frecker, Mary, and Noboru Kikuchi. "Optimal Synthesis of Compliant Mechanisms to Satisfy Kinematic and Structural Requirements Preliminary Results." Computers in Engineering Conference. Irvine: ASME, 1996. 1 9. [13] Kota, S., G. K. Ananthasuresh, S. B. Crary, and K. D. Wise. "Design and Fabrication of Microelectromechanical Systems." Journal of Mechanical Design 116 (December 1994): 1081 1088. [14] Howell, L. L., and A. Midha. "A Method for the Design of Compliant Mechanisms With Small Length Flexural P ivots." Transactions of the ASME 116 (March 1994): 280 290. [15] Howell, Larry. Compliant Mechanisms. New York: John Wiley & Sons, Inc., 2001. [16] Foulds, I., M. Trinh, S. Hu, S. Liao, R. Johnstone, and M. Parameswaran. "A Surface Micromachined Bistable Switch." Proceedings of the 2002 IEEE Canadian Conference on Electrical & Computer Engineering. IEEE, 2002. 465 469. [17] Opdahl, Patrick G., Brian D. Jensen, and Larry L. Howell. "An Investigation Into Compliant Bistable Mechanisms." Design Engineering Technical Conferences. Atlanta: ASME, 1998. 1 10. [18] Chiang, C. H. "Spherical Kinematics in Contrast to Planar Kinematics." Mechanism and Machine Theory 27, no. 3 (1992): 243 250. [19] Jagirdar, Saurabh. "Kinematics of Curved Flexible Beam." M.S. Thesis, University of South Florida, Tampa, 2006. [20] Le—n, Alejandro. "A Pseudo Rigid Body Model For Spherical Mechanisms: The Kinematics and Elasticity of a Curved Compliant Beam." M.S. Thesis, University of South Florida, Tampa, 2007. [21] Koester, D avid, Allen Cowen, Ramaswamy Mahadevan, Mark Stonefield, and Busbee Hardy. PolyMUMPs Design Handbook. 10. MEMSCAP, 2003. [22] Kolpekwar, Abhijeet, R. D. Blanton, and David Woodilla. "Failure Modes for Stiction in Surface Micromachined MEMS." International Test Conference. IEEE, 1998. 551 556.

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! 54 [23] Muller, Richard S. "MEMS: Quo Vadis in Century XXI." Microelectronic Engineering (Elsevier) 53 (2000): 47 54. [24] Alley, R. L., G. J. Cuan, R. T. Howe, and K. Komvopoulos. "The Effect of Release Etch Processing on Surface Microstructure Stiction." Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, 1992. [25] Bhushan, Bharat. Nanotribology and Nanomechanics of MEMS Devices. Department of Mechanical Engineering, The Ohio State University, Columbus: IEEE, 1996, 91 98. [26] Hestenes, David. New Foundations for Classical Mechanics. Edited by Alwyn Van Der Merwe. Dordrecht: D. Reidel Publishing Company, 1986. !

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! 55 APPENDICES

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! 56 Appendix A: ANSYS Code for a Restrained QBM !***************************************** !/INPUT,J: \ Thesis \ ANSYS \ QBM_restrained,txt,,1 !/CWD,'J: \ Thesis \ ANSYS' !***************************************** FINISH /CLEAR /FILENAME, Restrained Quadrantal Bistable Mechanism /title,Restrained Quadrantal Bistable Mechanism WRITE=1 1= Write output files, Else= Don't Write /PREP7 Enter the pre processor !***************************************** !********** Model Parameters ************* !***************************************** OP_Start=0 Theta for first case (multiple of 5) OP_Stop=90 Th eta for final case (multiple of 5) PI=acos( 1.) R=100 Radius of quadrantal beam h=2 Thickness b=5 Width of compliant segments !************ Define Area **************** A = h*b Cross sectional area of beams Iz= 1/12*b*h*h*h Second Moment of Area (aka Area Moment of Inertia) Ix= 1/12*h*b*b*b !***************************************** !********** Define Keypoi nts ************* !***************************************** Create Keypoints: K(Point #, X Coord, Y Coord, Z Coord) K,1,0,0,0, K,2,0, R,0, K,3,R,0,0,

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Appendix A (Continued) 57 !***************************************** !************ Create Links *************** !************* **************************** LARC,2,3,1,R, *Compliant* LESIZE,ALL,,,32 Divides the compliant arc into 32 segments !L,1,2,1 !******** Declare element type ********** ET,1,BEAM4 KEYOPT,1,2,1 KEYOPT,1,6,1 !******** Define Real Constants ********** R,1,A,Ix,Iz,h,b, Properties of compliant segments !****** Define Material Properties ****** MP,EX,1,170000 Young's Modulus of Elasticity (MPa) MP,PRXY,1,0.22 Poisson's ratio !***************************************** !**************** Mesh ******************* !***************************************** type,1 Use element type 1 mat,1 use material property set 1 real,1 Use real constant set 1 LMESH,ALL mesh all lines !******** Get Node # of keypoints ******** ksel,s,kp,,2 nslk,s *get,nkp2,node,0,num,max Re trieves and stores a value as a scalar or part of an array ksel,s,kp,,3 nslk,s *get,nkp3,node,0,num,max ALLSEL FINISH Finish pre processing !***************************************** !********** *** Solution ****************** !***************************************** *get,date,active,,dbase,ldate *get,time,active,,dbase,ltime year=nint(date/10000) month=nint(nint(date year*10000)/100) day=date (nint(date/100))*100

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Appendix A (Continued) 58 hour=nint(time/10000 .5) minu te=nint((time hour*10000)/100 .5) *DO,nn,OP_Start,OP_Stop,5 OPlanar_angle=nn !Theta Planar_angle =25 !Phi KEYW,PR_SGUI,1 Suppresses "Solution is Done" text box /SOL Enter the solution processor /gst,off Turn off graphical convergence monitor ANTYPE,0 Analysis type, static NLGEOM,1 Includes large deflection effects in a static or full tran sient analysis LNSRCH,AUTO AN SYS automatically switches line search on/off NEQIT,50 Set max # of iterations DELTIM ,,0.0001 Set minimum time step increment !***************************************** !**** Define Displacement Constraints***** !***************************************** DK,2,,0,,,UX,UY,UZ,ROTX,ROTY,ROTZ DK,3,,0,,,UX,UY,UZ,ROTX,ROTY,ROTZ !********* Displacement Load ************* LS1=OPlanar_angle *IF,LS1,GT,0,THEN If" statement used to avoid the warning when LS1=0 thus making the "Do loop" unnecessary *DO,step,1,LS1,1 theta=step*PI/180 disp2y= R*cos(theta)+R disp2z=R*sin(theta) DK,2,UY,disp2y DK,2,UZ,disp2z LSWRITE,step *ENDDO *ENDIF LS2=Planar_angle*2 *DO,step,1,LS2,1 phi=0.5*step*PI/180

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Appendix A (Continued) 59 disp3x=R*cos(phi) R disp3y=R*sin(phi) DK,3,UX,disp3x DK,3,UY,disp3y LSWRITE,LS1+step *ENDDO LS=LS1+LS2 LSSOLVE,1,LS FINISH Finish the solution processor !***************************************** !*********** Postprocessor *************** !*************************************** ** /POST1 Enter the postprocessor PLDISP,1 Displays deformed & undeformed shape /VIEW,1,1 Switches to right view /ANG,1, 90,ZS,0 Rotates View /ANG,1,15,XS,1 Rotates View SET,LAST /REPLOT *DIM,OPlanar,TABLE,LS,6 *DIM,Planar,TABLE,LS,6 *Do,i,1,LS SET,i Read data for step "i" *GET,fx1,Node,nkp2,RF,FX Assign Ortho planar data to OPlanar table *SET,OPlanar(i,1),fx1 *GET,fy1,Node,nkp2,RF,FY *SET,OPlanar(i,2),fy1 *GET,fz1,Node,nkp2,RF,FZ *SET,OPlanar(i,3),fz1 *GET,mx1,Node,nkp2,RF,MX *SET,OPlanar(i,4),mx1 *GET,my1,Node,nkp2,RF,MY *SET,OPlanar(i,5),my1 *GET,mz1,Node,nkp2,RF,MZ *SET,OPlanar(i,6),mz1 *GET,fx1,Node,nkp3,RF,FX Assign planar data to Planar table *SET,Planar(i,1),fx1 *GET,fy1,Node,nkp3,RF,FY *SET,Planar(i,2),fy1 *GET,fz1,Node,nkp3,RF,FZ *SET,Planar(i,3), fz1

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Appendix A (Continued) 60 *GET,mx1,Node,nkp3,RF,MX *SET,Planar(i,4),mx1 *GET,my1,Node,nkp3,RF,MY *SET,Planar(i,5),my1 *GET,mz1,Node,nkp3,RF,MZ *SET,Planar(i,6),mz1 *ENDDO *IF,WRITE,EQ,1,THEN *cfopen,J: \ Thesis \ ANSYS \ Results \ QBM_DATA_Theta=%OPlanar_angle%,te xt *vwrite,month,' ',day,' ',year,hour,':',minute %I %C %I %C %I %4.2I %C %2.2I *vwrite,'Note: The first',OPLanar_angle,'steps correspond to the movement','of the Ortho Planar link.','The final',LS2,'correspond to the planar link.' %C %I %C %C %/ %14C %I %C %/ *vwrite,'Planar:','angle phi = ',Planar_angle,'degrees' % 17C %C %I %C *vwrite,'FX','FY','FZ','MX','MY','MZ' % 17C % 17C % 17C % 16C % 16C % 16C *vwrite,Planar(1,1),Planar(1,2),Planar(1,3),Planar(1,4),Planar(1, 5),Planar(1,6) %16.8G %16.8G %16.8G %1 6.8G %16.8G %16.8G *vwrite,'Ortho Planar:','angle theta = ',OPlanar_angle,'degrees' %/ %/% 17C %C %I %C *vwrite,'FX','FY','FZ','MX','MY','MZ' % 17C % 17C % 17C % 16C % 16C % 16C *vwrite,OPlanar(1,1),OPlanar(1,2),OPlanar(1,3),OPlanar(1,4),OPlan ar(1,5),OPlanar(1,6) %16.8G %16.8G %16.8G %16.8G %16.8G %16.8G *cfclose *ENDIF FINISH *ENDDO

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Appendix A (Continued) 61 !***************************************** /POST1 Re enter the postpr ocessor following loops PLDISP,1 Displays deformed & undeformed shape /VIEW,1,1 Switches to right view /ANG,1, 90,ZS,0 Rotates Vie w /ANG,1,15,XS,1 Rotates View SET,LAST /REPLOT /FOC,1,AUTO,,,1 /FOC,1, 0.5, 0.05,,1 /REPLOT ANTIME,45,0.1, ,1,1,0,0 Animate

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! 62 Appendix B: ANSYS Code for an Unrestrained QBM !***************************************** !/INPUT,J: \ Thesis \ ANSYS \ QBM_unrestrained,txt,,1 !/CWD,'J: \ Thesis \ ANSYS' !***************************************** FINISH /CLEAR /FILENAME, Unrestrained Quadrantal Bistable Mechanism /title, Unrestrained Quadran tal Bistable Mechanism WRITE=1 1= Write output files, Else= Don't Write /PREP7 Enter the pre processor !***************************************** !********** Model Parameters ************* !***************************************** PI=acos( 1.) R=100 Radius of quadrantal beam h=2 Thickness b=5 Width of compliant segments !*********** Define Area **************** A = h*b Cross sectional area of beams Iz= 1/12*b*h*h*h Second Moment of Area (aka Area Moment of Inertia) Ix= 1/12*h*b*b*b !******************************* ********** !********** Define Keypoints ************* !***************************************** Create Keypoints: K(Point #, X Coord, Y Coord, Z Coord) K,1,0,0,0, K,2,0, R,0, K,3,R,0,0, K,4,R*cos( 45*PI/180),R*sin( 45*PI/180),0,

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Appendix B (Con tinued) 63 !***************************************** !************ Create Links *************** !***************************************** L,1,3,1 LARC,2,4,1,R, *Compliant* LARC,4,3,1,R, *Compliant* LESIZE,2,,,16 Divides the compliant arc into 16 segments LESIZE,3,,,16 Divides the compliant arc into 16 segments !******** Declare element type ********** ET,1,BEAM4 KEYOPT,1,2,1 KEYO PT,1,6,1 !******** Define Real Constants ********** R,1,A,Ix,Iz,h,b, Properties of compliant segments !****** Define Material Properties ****** MP,EX,1,170000 Young's Modulus of Elasticity ( MPa) MP,PRXY,1,0.22 Poisson's ratio !***************************************** !**************** Mesh ******************* !***************************************** type,1 Use element type 1 mat,1 use material property set 1 real,1 Use real constant set 1 LMESH,ALL mesh all lines !******** Get Node # of keypoints ******** ksel,s,kp,,1 nslk,s *get,nkp1,node,0,num,max Retrieves and stor es a value as a scalar or part of an array ksel,s,kp,,2 nslk,s *get,nkp2,node,0,num,max ksel,s,kp,,3 nslk,s *get,nkp3,node,0,num,max

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Appendix B (Con tinued) 64 ksel,s,kp,,4 nslk,s *get,nkp4,node,0,num,max ALLSEL FINISH Finish pre processing !***************************************** !************* Solution ****************** !***************************************** *get,date,active,,dbase,ldate *get,time,active,,dbase,ltime year=nint(date/10000) month=nint(nint(date year*10000)/100) day=date (nint(date/100))*100 hour=nint(time/10000 .5) minute=nint((time hour*10000)/100 .5) OPlanar_angle=90 !Theta KEYW,PR_SGUI,1 Suppresses "Solution is Done" text box /S OL Enter the solution processor /gst,off Turn off graphical convergence monitor ANTYPE,0 Analysis type, static NLGEOM,1 Include s large deflection effects in a static or full transient analysis LNSRCH,AUTO ANSYS automatically switches line search on/off NEQIT,50 Set max # of iterations DELTIM,,0.0001 Set minimum time step increment !***************************************** !**** Define Displacement Constraints***** !***************************************** DK,1,,0,,,UX,UY,UZ, ROTX,ROTY DK,3,,0,,,UZ

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Appendix B (Con tinued) 65 !********* Displacement Load ************* *DO,step,1,OPlanar_angle,1 theta=step*PI/180 disp2y= R*cos(theta)+R disp2z=R*sin(theta) DK,2,UY,disp2y DK,2,UZ,disp2z LSWRITE,step *ENDDO LSSOLVE,1,OPlanar_angle FINISH Finish the solution processor !***************************************** !*********** Postprocessor *************** !***************************************** /POST1 Enter the postprocessor *DIM,OPlanar,TABLE,OPlanar_ angle,6 *DIM,Planar,TABLE,OPlanar_angle,6 *DIM,Arc,TABLE,OPlanar_angle,6 *Do,i,1,OPlanar_angle SET,i Read data for step "i" *GET,fx1,Node,nkp2,RF,FX Assign Ortho planar data to OPlanar table *SET,OPlanar(i,1),fx1 *GET,fy1,Node,nkp2,RF,FY *SET,OPlanar(i,2),fy1 *GET,fz1,Node,nkp2,RF,FZ *SET,OPlanar(i,3),fz1 *GET,mx1,Node,nkp2,RF,MX *SET,OPlanar(i,4),mx1 *GET,my1,Node,nkp2,RF,MY *SET,OPlanar(i,5),my1 *GET,mz1,Node,nkp2,RF,MZ *SET,OPlanar(i,6),mz 1 *GET,x1,Node,nkp3,U,X Assign planar data to Planar table *SET,Planar(i,1),x1 *GET,y1,Node,nkp3,U,Y *SET,Planar(i,2),y1

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Appendix B (Con tinued) 66 *GET,z1,Node,nkp3,U,Z *SET,Planar(i,3),z1 *GET,fx1,Node,nkp3,RF,FX *SET,Planar(i,4),fx1 *GET,fy1,Node,nkp3,RF,FY *SET,Planar(i,5),fy1 *GET,fz1,Node,nkp3,RF,FZ *SET,Planar(i,6),fz1 *GET,x1,Node,nkp4,U,X Assign arc data & moment of origin to Arc table *SET,Arc(i,1),x1 *GET,y1,Node,nkp4,U,Y *SET,Arc(i,2 ),y1 *GET,z1,Node,nkp4,U,Z *SET,Arc(i,3),z1 *GET,mx1,Node,nkp1,RF,MX *SET,Arc(i,4),mx1 *GET,my1,Node,nkp1,RF,MY *SET,Arc(i,5),my1 *GET,mz1,Node,nkp1,RF,MZ *SET,Arc(i,6),mz1 *ENDDO *IF,WRITE,EQ,1,THEN *cfopen,E: \ Thesis \ ANSYS \ Results \ QBM_Unrestrained_data,text *vwrite,month,' ',day,' ',year,hour,':',minute %I %C %I %C %I %4.2I %C %2.2I *vwrite,'Planar:' %/ % 17C *vwrite,'Planar X','Planar Y','Planar Z','FX','FY','FZ' % 17C % 17C % 17C % 16C % 16C % 16 C *vwrite,Planar(1,1),Planar(1,2),Planar(1,3),Planar(1,4),Planar(1, 5),Planar(1,6) %16.8G %16.8G %16.8G %16.8G %16.8G %16.8G *vwrite,'Arc displacement & Moments at origin:' %/ %/ %C

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Appendix B (Con tinued) 67 *vwrite,'Arc X','Arc Y','Arc Z','O MX','O MY','O MZ' % 17C % 17C % 17C % 16C % 16C % 16C *vwrite,Arc(1,1),Arc(1,2),Arc(1,3),Arc(1,4),Arc(1,5),Arc(1,6) %16.8G %16.8G %16.8G %16.8G %16.8G %16.8G *vwrite,'Ortho Planar:' %/ %/% 17C *vwrite,'FX','FY','FZ','MX','MY','MZ' % 17C % 17C % 17C % 16C % 16C % 16C *vwrite,OPlanar(1,1),OPlanar(1,2),OPlanar(1,3),OPlanar(1,4),OPlan ar(1,5),OPlanar(1,6) %16.8G %16.8G %16.8G %16.8G %16.8G %16.8G *cfclose *ENDIF PLDISP,1 Displays deformed & undeformed shape /VIEW,1, 1 Switches to right view /ANG,1, 90,ZS,0 Rotates View /ANG,1,15,XS,1 Rotates View SET,LAST /REPLOT /FOC,1,AUTO,,,1 /FOC,1, 0.5, 0.05,,1 /REPLOT ANTIME,45,0.1, ,1,1,0,0 Animate

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! 68 Appendix C: MATLAB Code for a Restrained QBM % MATLAB analysis of restrained ANSYS FEA data clear all for Theta= 0:5:90; Phi = 5; Radius = 100; % **************Import ANSYS Data*************** if (Theta < 10) filename = sprintf( 'QBM_Data_Theta=%01d.text' ,Theta); else filename = sprintf( 'QBM_Data_Theta=%02d.text' ,Theta); end fileloc = 'J: \ Thesis \ ANSYS \ Results \ ; % Files located on flash drive (Check drive letter) fid = fopen([fileloc,filename]) ; % Opens the file rawfile = native2unicode(fread(fid))'; % Reads file as machine code & changes data to text fclose(fid); % Closes the data file Pointer1 = findstr( 'MZ' rawfile); % Finds end of first row of header titles Pointer2 = findstr( 'Ortho Planar' ,rawfile); % Finds start of second data set Pdata = str2num(rawfile(Pointer1(1)+3:Pointer2(2) 1)); OPdata = s tr2num(rawfile(Pointer1(2)+3:end)); % **************Define Rotations**************** OP_rot = [ [1:Theta]*pi/180, Theta*pi/180*ones([1 2*Phi])]; P_rot = [0*ones([1 Theta]),[.5:.5:Phi]*pi/180]; for i = 1:length(OP_rot) theta_step = OP_rot(i); phi_step = P_rot(i); % Rotation Matrices B = [1 0 0; 0 cos(theta_step) sin(theta_step); 0 sin(theta_step) cos(theta_step)]; A = [cos(phi_step) sin(phi_step) 0; sin(phi_step) cos(phi_step) 0; 0 0 1]; OP_F_o = OPdata(i, 1:3)'; % OP Force matrix in X,Y,Z frame OP_M_o = OPdata(i,4:6)';

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Appendix C (Continued) 69 % OP Moments matrix in X,Y,Z frame P_F_o = Pdata(i,1:3)'; % Planar Force matri x in X,Y,Z frame P_M_o = Pdata(i,4:6)'; % Planar Moments matrix in X,Y,Z frame OP_F_b(i,:) = (B*OP_F_o)'; % OP Force matrix in B frame OP_M_b(i,:) = (B*OP_M_o)'; % OP Moment matrix in B frame P_F_a(i,:) = (A*P_F_o)'; % Planar Force matrix in A frame P_M_a(i,:) = (A*P_M_o)'; % Planar Moment matrix in A frame S_OP_M(i,:) = OP_M_o(1,1) Radius*OP_F_b(i,3); % Total Moment about X for the OP link S_P_M(i,:) = P_M_o(3,1) + Radius*P_F_a(i,2); % Total Moment about Z for the P link end % *** ***********Plots & Figures***************** set(0, 'DefaultAxesColorOrder' ,[0.6,0.6,0.6]) figure(1) hold on order = 4; % Order of polynomial fit [p,S] = polyfit(P_rot(Theta+1:end)'*180/pi, S_OP_M(Thet a+1:end,1),order); [y,delta]= polyval(p,P_rot(Theta+1:end)'*180/pi,S); max_delta= max(delta); poly_roots = roots(p); [i,j,v] = find(imag(poly_roots)==0); real_roots = poly_roots(i); [i,j,v] = find(real_roots>0); positive_roots = real_roots(i); good_zero = min(positive_roots); plot(P_rot(Theta+1:end)*180/pi,S_OP_M(Theta+1:end,1), '*k' ) plot(P_rot(Theta+1:end)'*180/pi,y, 'k' ,P_rot(Theta+1:end)'*180/pi, y+delta,P_rot(Theta+1:end)'*180/pi,y delta) xlabel( 'Phi' 'FontSize' ,12) ylabel( 'Mx of OP link' 'FontSize' ,12) title( \ it{Phi vs. OP Moment}' 'FontSize' ,16)

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Appendix C (Continued) 70 figure(2) hold on [p,S] = polyfit(P_rot(Theta+1:end)'*180/pi, P_F_a(Theta+1:end,2),order); [y,delta]= polyval(p,P_rot(Theta+1:end)'*180/pi,S); plot(P_rot(Theta+1:end)*180/pi,P_F_a(Theta +1:end,2), '*k' ) plot(P_rot(Theta+1:end)*180/pi,y, 'k' ,P_rot(Theta+1:end)*180/pi,y+ delta,P_rot(Theta+1:end)*180/pi,y delta) xlabel( 'Phi' 'FontSize' ,12) ylabel( 'Planar Force' 'FontSize' ,12) title( \ it{Phi vs. Planar Force}' 'FontSize' ,16) if (Theta > 0) [P_force(Theta/5,:),P_delta] = polyval(p,good_zero,S); end figure(3) clf hold on plot3([0,1],[0,0],[0,0]) plot3([0,0],[0, 1],[0,0]) plot3([0,cos(Phi*pi/180)],[0,sin(Phi*pi/180)],[0,0], 'k' ) plot3([0,0],[0, cos(Theta*pi/180)],[0,sin(Theta*pi/180)], 'k' ) x1 = [cos(P_rot(Theta+1:end)),zeros(size([1:2*Phi]))]; y1 = [sin(P_rot(Theta+1:end)), cos(Theta*pi/180)*ones(size([1:2*Phi]))]; z1 = [zeros(size([1:2*Phi])),sin(Theta*pi/180)*ones(size([1:2*Phi]))]; u1 = [Pdata(Theta+1:Theta + 2* Phi,1),OPdata(Theta+1:The ta + 2* Phi,1)]; v1 = [Pdata(Theta+1:Theta + 2* Phi,2),OPdata(Theta+1:Theta + 2* Phi,2)]; w1 = [Pdata(Theta+1:Theta + 2* Phi,3),OPdata(Theta+1:Theta + 2* Phi,3)]; quiver3(x1,y1,z1, u1, v1, w1,5, 'r' ) view(3) xlabel( 'X' 'FontSize' ,12) ylabel( 'Y' 'FontSiz e' ,12) zlabel( 'Z' 'FontSize' ,12) title( \ it{Force Vectors on Links}' 'FontSize' ,16)

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Appendix C (Continued) 71 axis equal grid on figure(4) hold on Px = [Radius*cos(P_rot(Theta+1:end))]; Py = [Radius*sin(P_rot(Theta+1:end))]; Pz = [zeros(size([1:2*Phi]))]; OPx = [zeros(size([1:2*Phi]))]; OPy = [Radius* cos(Theta*pi/180)*ones(size([1:2*Phi]))]; OPz = [Radius*sin(Theta*pi/180)*ones(size([1:2*Phi]))]; Dist_AB = sqrt((OPx Px).^2+(OPy Py).^2+(OPz Pz).^2); D_init = Radius sqrt(2); D_change = Dist_AB D_init; OP Fx = [OPdata(Theta+1:Theta + 2* Phi,1)]; OPFy = [OPdata(Theta+1:Theta + 2* Phi,2)]; OPFz = [OPdata(Theta+1:Theta + 2* Phi,3)]; OPF_mag = sqrt(OPFx.^2 + OPFy.^2 + OPFz.^2); PFx = [Pdata(Theta+1:Theta + 2* Phi,1)]; PFy = [Pdata(Theta+1:Theta + 2* Phi,2)]; PFz = [Pdata(Theta+1:Theta + 2* Phi,3)]; PF_mag = sqrt(PFx.^2 + PFy.^2 + PFz.^2); Avg_F = (OPF_mag + PF_mag)/2; K = Avg_F./ D_change'; plot(D_change,Avg_F, *k' ) xlabel( 'Change in Length' 'FontSize' ,12) ylabel( 'Force' 'FontSize' ,12) title( \ it{Change in Length vs. Force}' 'FontSize' ,16)

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Appendix C (Continued) 72 figure(5) hold on Phi_change = P_rot(Theta+1:end)*180/pi; plot(Phi_change,S_P_M(Theta+1:end), *k' ) xlabel( 'Phi' 'FontSize' ,12) ylabel( 'Mz of P link' 'FontSize' ,12) title( \ it{Phi vs. Planar Mo ment}' 'FontSize' ,16) %pause end figure(6) hold on [p,S] = polyfit([5:5:Theta]',P_force(1:end),order); [y,delta]= polyval(p,[5:5:Theta]',S); max(delta) plot([5:5:Theta],P_force(1:end), 'dk' ) plot([5:5:Theta]',y, 'k' ,[5:5:Theta]',y+delta,[5:5:Theta]',y delta) xlabel( 'Out of plane angle, \ theta' 'FontSize' ,12) ylabel( 'Planar Threshold Force ( \ mu N)' 'FontSize' ,12) title( \ it{Theta vs. Planar Threshold Force}' 'FontSize' ,16)

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! 73 Appendix D: MATLAB Code for an Unrestrained QBM % MATLAB analysis of unrestrained ANSYS FEA data clear all Theta = 90; Radius = 100; % **************Import ANSYS Data*************** filename = 'QBM_Unrestrained_data.text' ; fileloc = 'J: \ Thesis \ ANSYS \ Results \ ; % Files located on flash drive (Check drive letter) fid = fopen([fileloc,filename]); % Opens the file rawfile = native2unicode(fread(fid))'; % Reads file as ma chine code & changes data to text fclose(fid); % Closes the data file Pointer1 = findstr( 'FZ' rawfile); % Finds end of first row of header titles Pointer2 = findstr( 'Arc position' ,r awfile); % Finds start of second data set Pointer3 = findstr( 'MZ' ,rawfile); % Finds end of second & third row of header titles Pointer4 = findstr( 'Ortho Planar' ,rawfile); % Finds start of third data set Pdata = str2num(rawfile(Pointer1(1)+3:Pointer2(1) 1)); Arc_Origin_data = str2num(rawfile(Pointer3(1)+3:Pointer4(1) 1)); OPdata = str2num(rawfile(Pointer3(2)+3:end)); % **************Define Rotations**************** OP_Loc = [zeros([1 Theta]); Radius*cos([1:Theta]*pi/180);Radius*sin([1:Theta]*pi/180)]'; for i = 1:1:Theta % Rotation Matrix for OP link B = [1 0 0; 0 cos(i*pi/180) sin(i*pi/180); 0 sin(i*pi/180) cos(i*pi/180)]; P_Loc(i,:) = ([Radius,0,0] + Pdata(i ,1:3))'; % Planar Location matrix in X,Y,Z frame P_F(i,:) = Pdata(i,4:6)'; % Planar Force matrix in X,Y,Z frame Arc_Loc(i,:) = ([Radius*cos(45*pi/180), Radius*sin(45*pi/180),0] + Arc_Origin_da ta(i,1:3))'; Origin_M(i,:)= Arc_Origin_data(i,4:6)'; % Moments at the origin matrix in X,Y,Z frame

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Appendix D (Continued) 74 OP_F(i,:) = OPdata(i,1:3)'; % OP Force matrix in X,Y,Z frame OP_M(i,:) = OPdata( i,4:6)'; % OP Moments matrix in X,Y,Z frame % **** compare with Alex's model ***** OP_F_B(i,:) = B*OPdata(i,1:3)'; % OP Force matrix in B frame OP_M_B(i,:) = B*OPdata(i,4:6)'; % OP Moments matrix in B frame Arc_angle_s=atan2(Arc_Loc(i,2),Arc_Loc(i,1))*180/pi; C(i,:,:) = [cos(Arc_angle_s) sin(Arc_angle_s) 0; sin(Arc_angle_s) cos(Arc_angle_s) 0; 0 0 1]; end % **************Plots & Figures***************** set(0, 'DefaultAxesColorOrder' ,[0.6,0.6,0.6]) figure(1) clf Dis1 = P_Loc Arc_Loc; Dis2 = OP_Loc Arc_Loc; D1= sqrt(Dis1(:,1).^2+Dis1(:,2).^2+Dis1(:,3).^2); D2= sqrt(Dis2(:,1).^2+Dis2(:,2).^2 +Dis2(:,3).^2); plot([1:Theta],D1, *k' ,[1:Theta],D2, *g' ) xlabel( 'Theta' 'FontSize' ,12) ylabel( 'Distance to Arc Center' 'FontSize' ,12) title( \ it{Distance to Arc Center}' 'FontSize' ,16) legend( 'Planar link' 'OP link' 'Location' 'SouthWest' ) figure(2) clf plot([1:Theta],D1 D2, '*k' ) xlabel( 'Theta' 'FontSize' ,12) ylabel( 'Change in Distance' 'FontSize' ,12) title( \ it{Change in Distance}' 'FontSize' ,16) figure(3) clf PR =sqrt(P_Loc(:,1).^2+P_Loc(:,2).^2+P_Loc(:,3).^2); OPR =sqrt(OP_Loc( :,1).^2+OP_Loc(:,2).^2+OP_Loc(:,3).^2); ArcR =sqrt(Arc_Loc(:,1).^2+Arc_Loc(:,2).^2+Arc_Loc(:,3).^2); plot([1:Theta],PR, 'b' ,[1:Theta],OPR, 'r' ,[1:Theta],ArcR, 'g' )

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Appendix D (Continued) 75 xlabel( 'Theta' 'FontSize' ,12) ylabel( 'Radial Distance' 'FontSize' ,12) title( \ it{Radial Distance}' 'FontSize' ,16) legend( 'Planar link' 'OP link' 'Arc Center' 'Location' 'SouthWest' ) figure(4) clf Arc_angle1=atan2(Arc_Loc(:,2),Arc_Loc(:,1))*180/pi; Arc_angle2=atan2(Arc_Loc(:,3),sqrt(Arc_Loc(:,1).^2+Arc_Loc(:,2).^ 2))*180/pi; P_angle1=atan2(P_Loc(:,2),P_Loc(:,1)); P_angle2=atan2(P_Loc(:,3),sqrt(P_Loc(:,1).^2+P_Loc(:,2).^2)); OP_angle1=atan2(OP_Loc(:,2),OP_Loc(:,1)); OP_angle2=atan2(OP_Loc(:,3),sqrt(OP_Loc(:,1).^2+OP_Loc(:,2).^2)); Ang1 = (OP_angle1+P_angle1)/2*180/pi; Ang2 = (OP_angle2+P_angle2)/2*180/pi; plot([1:Theta],Arc_angle1, *g' ,[1:Theta],Ang1, *b' ) figure(5) clf plot([1:Theta],Arc_angle2, *g' ,[1:Theta],Ang2, *b' ) % The green line is significant only because it shows the inward % sagging of the compliant link, due in part to tensile forces. % It also shows a violation of t he assumption of Alex/Saurabh's % model, though not a large one. figure(6) clf hold on quiver3(OP_Loc(:,1),OP_Loc(:,2),OP_Loc(:,3),OP_F(:,1),OP_F(:,2),O P_F(:,3)) quiver3(OP_Loc(:,1),OP_Loc(:,2),OP_Loc(:,3),OP_M(:,1),OP_M(:,2),O P_M(:,3), 'k' ) OP_M_mag = (OP_M(:,1).^2+OP_M(:,2).^2+OP_M(:,3).^2).^0.5; OP_F_mag = (OP_F(:,1).^2+OP_F(:,2).^2+OP_F(:,3).^2).^0.5; OP_F_unit = OP_F./(OP_F_mag*[1 1 1]); M_par_mag = dot(OP_F_ unit',OP_M')'*[1 1 1]; M_par = M_par_mag.*OP_F_unit; M_perp = OP_M M_par; M_perp_mag = (M_perp(:,1).^2+M_perp(:,2).^2+M_perp(:,3).^2).^0.5; M_perp_unit = M_perp./(M_perp_mag*[1 1 1]); d_unit = cross(OP_F_unit',M_perp_unit')'; d_mag1 = M_perp_mag./OP_F_m ag; d1 = (d_mag1*[1 1 1]).*d_unit;

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Appendix D (Continued) 76 %quiver3(OP_Loc(:,1),OP_Loc(:,2),OP_Loc(:,3),d1(:,1),d1(:,2),d1(: ,3),0,'g') screwpt1 = OP_Loc+d1; % center calculations quiver3(Arc_Loc(:,1),Arc_Loc(:,2),Arc_Loc(:,3),OP_F(:,1),OP_F(:,2 ),OP_F(:,3), 'b' ) M_center = cross( Dis2,OP_F)+OP_M; quiver3(Arc_Loc(:,1),Arc_Loc(:,2),Arc_Loc(:,3),M_center(:,1),M_ce nter(:,2),M_center(:,3), 'g' ) M_C= (squeeze(C(i,:,:))*M_center')'; F_C= (squeeze(C(i,:,:))*OPdata(:,1:3)')'; M_center_mag = (M_center(:,1).^2+M_center(:,2).^2+M_c enter(:,3).^2).^0.5; %OP_F_mag = (OP_F(:,1).^2+OP_F(:,2).^2+OP_F(:,3).^2).^0.5; %OP_F_unit = OP_F./(OP_F_mag*[1 1 1]); M_center_par_mag = dot(OP_F_unit',M_center')'*[1 1 1]; M_center_par = M_center_par_mag.*OP_F_unit; M_center_perp = M_center M_center_ par; M_center_perp_mag = (M_center_perp(:,1).^2+M_center_perp(:,2).^2+M_center_perp(:,3).^ 2).^0.5; M_center_perp_unit = M_center_perp./(M_center_perp_mag*[1 1 1]); d_unit2 = cross(OP_F_unit',M_center_perp_unit')'; d_mag2 = M_center_perp_mag./OP_F_mag; d2 = (d_mag2*[1 1 1]).*d_unit2; %quiver3(Arc_Loc(:,1),Arc_Loc(:,2),Arc_Loc(:,3),d2(:,1),d2(:,2),d 2(:,3),0,'g' ) screwpt2 = Arc_Loc+d2; screw_axis = screwpt2 screwpt1; check = cross(screw_axis',OP_F')'; quiver3(screwpt1(:,1),screwpt1(:,2),screwpt1(:,3),scr ew_axis(:,1) ,screw_axis(:,2),screw_axis(:,3),0, 'r' ) %plot([1:Theta],M_center,' *') xlabel( 'X' 'FontSize' ,12) ylabel( 'Y' 'FontSize' ,12) zlabel( 'Z' 'FontSize' ,12) title( \ it{Equivalent Screwpoint of Arc Center}' 'FontSize' ,16) legend( 'OP Forces' 'OP Moments' 'Arc Center Forces' 'Arc Center Moments' 'Screwpoint' 'Location' 'Best' ) grid on view(38,27)

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Appendix D (Continued) 77 % **************Half PRBM*********************** % Leon PRBM: Use two 45 degree models % Asp = height/width = 2/5 = 0.4 Beta = [0.5:.5:45]'*pi/1 80; %lambda = 45*pi/180*(1+2*[.1:.45]); % possible change in lambda due to elongation lambda = 45*pi/180; gamma = 0.846; %gamma = 1.25; c_theta = 1.21; % parametric angle coefficient K_f = 1.9; K_m = 0.2; Cap_Theta = asin(sin(Beta)/sin(gamma*lambda)); flag1 = 1 abs(sign(imag(Cap_Theta))); Cap_Theta = Cap_Theta./flag1; Phi = atan2(1 cos(Cap_Theta),cot(gamma*lambda)+cos(Cap_Theta)*tan(gamma*lambda) ); Alpha = pi/4 Phi; h=2; b=5; E=17 0000; I=1/12*b*h*h*h; DTB=cos(Cap_Theta).*sin(gamma*lambda)./cos(Beta); F_model = K_f*(E*I/(lambda*Radius^2))*Cap_Theta.*DTB; M_model = K_m*(E*I/(lambda*Radius))*Cap_Theta.*DTB; figure(4) % add to figur e 4 hold on plot(2*Beta*180/pi, 45 Phi*180/pi, 'yo ) xlabel( 'OP Angle, \ theta' 'FontSize' ,12) ylabel( 'Planar Angle of Arc Center' 'FontSize' ,12) title( \ it{Planar Angles of the Arc Center}' 'FontSize' ,16) legend( 'Planar angle of arc center' 'Average planar angle of P & OP links' 'Planar angle of Half PRBM' 'Location' 'SouthWest' )

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Appendix D (Continued) 78 figure(5) % add to figure 5 hold on plot(2*Beta*180/pi,Beta*180/pi, 'yo ) xlabel( 'OP Angle, \ theta' 'FontSize' ,12) ylabel( 'OP Angle of Arc Center' 'FontSize' ,12) title( \ it{Ortho Planar Angles of the Arc Center}' 'FontSize' ,16) legend( 'OP angle of arc center' 'Average OP angle of P & OP links' 'OP angle of Half PRBM' 'Location' 'SouthEast' ) figure(7) clf hold on plot(2*Beta*180/pi,F_C(:,3), 'b* ) plot(2*Beta*180/pi,F_model, 'k* ) xlabel( 'Theta' 'FontSize' ,12) ylabel( 'Force' 'FontSize' ,12) title( \ it{Forces}' 'FontSize' ,16) legend( 'Actual' 'Predicted by model' 'Location' 'Best' ) figure(8) clf hold on plot(2*Beta*180/pi,M_C(:,2), 'b* ) plot(2*Beta*180/pi,M_model, 'k* ) xlabel( 'Theta' 'FontSize' ,12) ylabel( 'Moment' 'FontSize' ,12) title( \ it{Moments}' 'FontSize' ,16) legend( 'Actual' 'Predicted by model' 'Location' 'Best' ) figure(9) clf hold on plot(2*Beta*180/pi,M_C(:,2)./F_C(:,3), 'b* ) plot(2*Beta*180/pi,M_model./F_model, 'k* ) xlabel( 'Theta' 'FontSize' ,12) ylabel( 'Moment/Force' 'FontSize' ,12) title( \ it{Ratio of Moments/Forces}' 'FontSize' ,16) legend( 'Actual' 'Predicted by model' 'Location' 'Best' )

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! 79 Appendix E: ANSYS Code for the BAP Figure E 1 ANSYS Model of the BAP !************************************************************** !/CWD,'J: \ Thesis \ ANSYS' !************************************************************** FINISH /CLEAR /FILNAM,Bistable Aerial Platform /title,Bistable Aerial Platform /PREP7 Enter the pre processor

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Appendix E (Continued) 80 !**************************** ************ !********** Model Parameters ************ !**************************************** PI=acos( 1.) R1=76 Radius of elevating ring R2=91.5 Radius of input link (a ka actuator) R3=99.5 Radius of bistable element R4=137 Radius of guide ring L=196 Total length of ortho planar beam phi= 120*PI/180 Initial angle of position for the input link psi= 260*PI/180 Initial angle to define the connection between ortho planar beam and bistable element h1=2 Thickness of Poly1 h2=1.5 Thickness of Poly2 b1=42 Width of ortho planar beam b2=10 Width of remaining beam elements b3=20 Width of elevating ring b4=5 Width of compliant segments (actuator and bistable element) !************ Define Area *************** A1 = h2*b1 Cross sectional area of the ortho planar beam Iz1= 1/12 *b1*h2*h2*h2 Second Moment of Area (aka Area Moment of Inertia) Ix1= 1/12*h2*b1*b1*b1 A2= h1*b2 Cross sectional of the remaining beam elements Iz2= 1/12*b2*h1* h1*h1 Ix2= 1/12*h1*b2*b2*b2 A3= h1*b3 Cross sectional area of the elevating ring Iz3= 1/12*b3*h1*h1*h1 Ix3= 1/12*h1*b3*b3*b3 A4= h1*b4 Cross sectional area of the compliant segments

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Appendix E (Continued) 81 Iz4= 1/12*b4*h1*h1*h1 Ix4= 1/12*h1*b4*b4*b4 !***************************************** !********** Define Keypoints ************* !***************************************** Create Keypoints: K(Point # X Coord, Y Coord, Z Coord) Point of rotation for ortho planar link K,1,0,0,h1+h2/2, K,2,0,0,h1+h2/2, Actuation points K,3,R2*cos(phi),R2*sin(phi),0, K,4,R2*cos(phi+PI),R2*sin(phi+PI),0, Center of actuator link K,5,R2*cos(phi/2),R2*sin(phi/2),h1 *2, K,6, R2*cos(phi/2), R2*sin(phi/2),h1*2, Rotating beams K,7,R2,0,0, K,8, R2,0,0, K,9,R3,0,0, K,10, R3,0,0, Point of connection between the ortho planar link and the quadrantal beam K,11,R3*cos(psi),R3*sin(psi),h1+h2/2, K,12, R3*cos(psi), R3*sin(psi),h1+h2/2, Elevating Ring K,13,R1*cos(phi/2),R1*sin(phi/2),0, K,14, R1*cos(phi/2), R1*sin(phi/2),0, K,15,R1*cos(phi/2+PI/2),R1*sin(phi/2+PI/2),0, K,16,R1*cos(phi/2 PI/2),R1*sin(phi/2 PI/2),0, Points on ortho planar link K ,17,0,R3*sin(psi),h1+h2/2, K,18,0, R3*sin(psi),h1+h2/2, K,19,0, L,h1, K,20,0,L,h1,

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Appendix E (Continued) 82 Additional points at origin K,21,0,0,0, K,22,0,0,0, Guide "Ring" K,23,R4,0,0, K,24, R4,0,0 !***************************************** !************ Create Links *************** !***************************************** !*********** Straight links ************** Ortho planar links LSTR, 1, 17 #1 LSTR, 2, 18 #2 LSTR, 17, 11 #3 LSTR, 18, 12 #4 LSTR, 17, 19 #5 LSTR, 18, 20 #6 Beam connecting actuations points LSTR, 3, 21 #7 LSTR, 21, 4 #8 Rotating Beams LSTR, 7, 9 #9 LSTR, 8, 10 #10 LSTR, 22, 23 #11 LSTR, 22, 24 #12 LSTR, 23, 9 #13 LSTR, 24, 10 #14 Beam attaching center of actuator arc to elevating ring LSTR, 5, 13 #15 LSTR, 6, 14 #16 LESIZE,ALL,,,1 Specifies a single division on unmeshed lines LESIZE,1,,,3 LESIZE,2,,,3 LESIZE,5,,,3 LESIZE,6,,,3

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Appendix E (Continued) 83 !************** Arc links **************** Elevat ing ring LARC,16,13,21,R1, #17 LARC,14,16,21,R1, #18 LARC,15,13,21,R1, #19 LARC,14,15,21,R1, #20 Actuator LARC,3,5,21,R2, #21 *Compliant* LARC,5,7,21,R2, #22 *Compli ant* LARC,4,6,21,R2, #23 *Compliant* LARC,6,8,21,R2, #24 *Compliant* Quadrantal Beam LARC,9,11,21,R3, #25 *Compliant* LARC,10,12,21,R3, #26 *Compliant* Guide Ring !LARC,23,24,22,R4, #27 !LARC,24,25,22,R4, #28 !LARC,25,26,22,R4, #29 !LARC,26,23,22,R4, #30 LSEL,S,,,17,20 Divides the rigid arcs into 8 segments !LSEL,A,,,27,30 LESIZE,ALL,,,8 LSEL,S,,,21,24 Actuator divided into two parts with 16 segments each LESIZE,ALL,,,16 LSEL,S,,,25,26 Divides the compliant arcs of the quadrantal beams into 32 segments LESIZE,ALL,,,32 LSEL,ALL !******** Declare eleme nt type ********** ET,1,BEAM4 KEYOPT,1,2,1 KEYOPT,1,6,1 !******** Define Real Constants ********* R,1,A1,Ix1,Iz1,h2,b1, Properties of ortho planar beam R,2,A2,Ix2,Iz2,h1,b2, Properties of other beam elements R,3,A3,Ix3,Iz3,h1,b3, Properties of elevating ring R,4,A3,Ix4,Iz4,h1,b4, Properties of compliant segments

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Appendix E (Continued) 84 !****** Define Material Properties ****** MP,EX,1,170000 Young's Modulus of Elasticity (MPa) MP,PRXY,1,0.22 Poisson's ratio !***************************************** !**************** Mesh ******************* !***************************************** type,1 Use element type 1 mat,1 Use materia l property set 1 real,1 Use real constant set 1 LMESH,1,8 Mesh lines 1 8 !LMESH,27,30 Mesh lines 27 30 real,2 Use real constant set 2 LMESH,9,16 mesh lines 9 16 real,3 Use real constant set 3 LMESH,17,20 Mesh lines 17 20 real,4 Use real constant set 4 LMESH,21,26 Mesh lines 21 26 !***************************************** !******** Create Contact Elements ******** !***************************************** ET,2,Targe170 ET,3,Conta176 KEYOPT,3,2,1 Use penalty fun c tion contact algor ithms KEYOPT,3,3,1 Specify crossing beams KEYOPT,3,5,3 Reduce gap/penetration with auto CNOF KEYOPT,3,6,1 Allow nominal refinement to contact stiffness KEYOPT,3,7,3 Allow change in the contact predictions KEYOPT,3,10,1 Update contact stiffness after each substep

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Appendix E (Continued) 85 R,5,h2/2,h1/2,0.25 R,6,h2/2,h1/2,0.25 REAL,5 Generate the target surface LSEL,S,,,1 LS EL,A,,,5 TYPE,2 NSLL,S,1 ESLL,S,0 ESURF ESURF,,REVE Generate the contact surface LSEL,S,,,17,18 TYPE,3 NSLL,S,1 ESLL,S,0 ESURF,,BOTTOM REAL,6 Generate the target surface LSEL,S,,,2 LSEL,A,,,6 TYPE,2 NSLL,S,1 ESLL,S,0 ESURF ESURF,,REVE Generate the contact surface LSEL,S,,,19,20 TYPE,3 NSLL,S,1 ESLL,S,0 ESURF,,TOP ALLSEL !***************************************** !******** Get Node # of Keypoints ******** !***************************************** ksel,s,kp,,3 nslk,s *get,nkp3,node,0,num,max

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Appendix E (Continued) 86 ksel,s,kp,,4 nslk,s *get,nkp4,node,0,num,max ksel,s,kp,,5 nslk,s *get,nkp5,node,0,num,max ksel,s,kp,,6 nslk,s *get,nkp6,node,0,num,max ksel,s,kp,,7 nslk,s *get,nkp7,node,0,num,max ksel,s,kp,,8 nslk,s *get,nkp8,node,0,num,max ksel,s,kp,,11 nslk,s *get,nkp11,node,0,num,max ksel,s,kp,,12 nslk,s *get,nkp12,node,0,num,max ksel,s,kp,,22 nslk,s *get,nkp22,node,0,num,max ALLSEL FINISH Finish pre processing !***************************************** !****** ******* Solution ****************** !***************************************** KEYW,PR_SGUI,1 Suppresses "Solution is Done" text box /SOL Enter the solution processor /gst,off Turn off the g raphical convergence monitor

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Appendix E (Continued) 87 ANTYPE,0 Analysis type, static NLGEOM,ON Includes large deflection effects in a static or full transient analysis SOLCONTROL,ON,ON Gives ANSYS control of the time step size LNSRCH,AUTO ANSYS automatically switches line search on/off NEQIT,100 Set max # of iterations to 100 DELTIM,,0.0001 Set minimum time step increment !**** Define Displacement Constraints***** DK,1,,0,,,UX,UY,UZ,ROTY,ROTZ DK,2,,0,, ,UX,UY,UZ,ROTY,ROTZ DK,3,UZ,0 DK,4,UZ,0 !DK,9,UZ,0 !DK,10,UZ,0 KSEL,S,,,23,24 DK,ALL,,0,,,UZ KSEL,S,,,13,16 DK,ALL,,0,,,ROTX,ROTY KSEL,S,,,21,22 DK,ALL,,0,,,UX,UY,UZ,ROTY,ROTX KSEL,ALL !********* Displacement Load ************* !ESEL,S,TYPE,,2,3, Select element types 2 & 3 which make up the contact pairs !EKILL,ALL Kill all selected elements !ESEL,ALL Re select all elements CNCHECK,ADJUST Mo ve contact nodes to close gap or reduce penetration loadsteps=300 *DO,step,1,loadsteps,1 theta=0.25*step*PI/180 DK,21,ROTZ,theta LSWRITE,step *ENDDO

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Appendix E (Continued) 88 !LSSOLVE,1,loadsteps theta1=ROTZ(nkp22) LD=loadsteps *DO,step,1,10,1 DK,22,ROTZ,theta1 + 2*step*PI/180 LSWRITE,LD+step loadsteps=loadsteps+1 *ENDDO LSSOLVE,1,loadsteps KEYW,PR_SGUI,0 Undo suppression of "Solution is Done" text box FINISH Finish the solution processor !***************************************** !*********** Postprocessor *************** !***************************************** /POST1 Enter the postprocessor PLDISP,1 Displays deformed & undeformed shape !/VIEW,1, 1 Switches to left view /VIEW,1,, 1 Switches to bottom view /ANG,1,30,YS,1 Rotates View /REPLOT,FAST ANTIME,50,0.05, ,1,1,0,0 Animate

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! 89 Appendix F: Coordinate Transformation The ANSYS data taken provides the forces experienced by the model in terms of the original xyz coordinate system. However, in order to properly monitor the direc tion of the forces at the ends of the rotating links, a new coordinate system placed at the end of each link is needed. To accomplish this, the forces must be transformed into new frames of reference. The method used to accomplish the transformation is exp lained below. F.1. Mathematics of Rotations A vector x is composed of both a parallel component x || and a perpendicular component x # with respect to a chosen plane [26] ( F. 1 ) Linearly transforming x by applying a rotation about a vector, A which is normal to the plane, will result in ( F. 2 ) where x || e i A describes a rotation of x || in the plane about A and through an angle of magnitude | A | [26] Note that this rotation only affects x || and that x # = x $ # [26] The resulting transformation is depicted in Figure F 1

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Appendix F (Continu ed) 90 Figure F 1 Rotation of a vector [ 2 6 ] 6 The rotation, R resulting from the application of e i A proves more useful when expressed in the parametric form ( F. 3 a) ( F. 3b) And since ( F. 4 ) the parameters % & and A can be related by ( F. 5 ) ( F. 6 ) where is a unit vector along the axis of rotation, A In practice, however, a rotation is typically expressed as !!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 6 Image adapted from Hestenes

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Appendix F (Continu ed) 91 ( F. 7 ) where = and represents the magnitude of the rotation, |A| which indicates that A = Thus, equations ( F. 5 ) and ( F. 6 ) respectively become ( F. 8 ) ( F. 9 ) Combining these equations, we arrive at ( F. 10 a) ( F. 10b) ( F. 10c) which will be used to develop rotation matrices for the ANSYS model. F.2. Rotation Matrices Using the equations developed in the previous section, the data taken from ANSYS can now be transformed from the original frame of reference, O, into new frames, A and B, that rotate with the planar and ortho planar links, respectively. The original and moving frames are shown below in Figure F 2 Starting with the ortho

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Appendix F (Continu ed) 92 Figure F 2 Rotating frames on the qu adrantal beam planar link and using equation ( F. 10 a), the rotated B frame becomes ( F. 11 ) because, as shown in Figure F 2 the B frame results from a rotation of the original frame about the x axis by a degree Theta, Likewise, the rotated A frame represents a rotation about the original z axis by a degree Phi, and becomes ( F. 12 )

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Appendix F (Continu ed) 93 Expan ding equation ( F. 11 ) by using equations ( F. 10 b) for the x axis results in ( F. 13 a) ( F. 13b) In this instance, since the axis of interest is also the axis of rotation, there is no B || because O x is tangent to the plane of rotation. Therefore ( F. 14 ) Such a simplification results from properly selecting the location and orientation of the new coordinate system; also note that because of this choice, O y and O z lie parallel to the plane of rotation for the B frame. For y axis, expanding equation ( F. 11 ) by using equations ( F. 10 c) yields ( F. 15 a) ( F. 15b) Simplifying this equation requires the understanding that ( F. 16 a) ( F. 16b) ( F. 16c) and that ( F. 17 a) ( F. 17b) where both j and k can represent any one of the three axes. Thus equation ( F. 15 b) becomes

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Appendix F (Continu ed) 94 ( F. 18 ) Since the component of interest, O y lies parallel to the plane of rotation for the B frame as stated before, B # =0 and B || =O y which will give a final result of ( F. 19 ) Similarly, it can be found that ( F. 20 ) Combining equations ( F. 14 ) ( F. 19 ) and ( F. 20 ) into a matrix yields ( F. 21 ) which, if viewed as ( F. 22 ) indicates that the original rotation, R(O x ), applied in ( F. 11 ) is equal to ( F. 23 ) Applying these same methods to equation ( F. 12 ) for the A frame will show that ( F. 24 ) and that

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Appendix F (Continu ed) 95 ( F. 25 ) With equations ( F. 23 ) and ( F. 25 ) now developed, the force data provided by ANSYS can be adjusted for the planar and ortho planar links, allowing F x F y and F z to be seen in terms of the relative position and orientation of the link.