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Modeling and parameter study of bistable spherical compliant mechanisms

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Modeling and parameter study of bistable spherical compliant mechanisms
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Smith, Chester
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University of South Florida
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3d Mems
Microelectromechanical
Ortho-planar
Virtual Work
Dissertations, Academic -- Engineering Mechanical Engineering -- Masters -- USF   ( lcsh )
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bibliography   ( marcgt )
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Abstract:
ABSTRACT: The bistable spherical compliant mechanism (BSCM) is a novel device capable of large, repeatable, out-of-plane motion, characteristics that are somewhat difficult to achieve with surface micromachined microelectromechanical systems. An improved pseudo-rigid-body model (PRBM) to predict the behavior of the BSCM is presented. The new model was used to analyze seven different versions of the device, each with a different compliant joint length. The new model, which adds torsion, is compared with a finite element analysis (FEA) beam model. The new model more closely approximates the results yielded by FEA than previous models used to analyze the BSCM. Future work is needed to quantify stress-stiffening interactions between bending and torsion. Both FEA and the current models show that increasing the length of the compliant segment decreases the amount of force required to actuate the device.
Thesis:
Thesis (M.S.M.E.)--University of South Florida, 2011.
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Includes bibliographical references.
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by Chester Smith.
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Modeling and Parameter Study of Bistable Spherical Compliant Mechanisms by Chester Law Smith 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 Lusk, Ph.D. Autar Kaw, Ph.D. Kyle Reed, Ph.D. Date of Approval: March 11, 2011 Keywords: Ortho-Planar, Microelectromechanical System, 3D MEMS, Vi rtual Work, Static Jump Copyright 2011, Chester Law Smith

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i Table of Contents List of Tables iii List of Figures iv Abstract vi Chapter 1: Introduction 1 1.1 The Bistable Spherical Compliant Mechanism 2 Chapter 2: Outline 4 2.1 Objective 4 2.2 Motivation 4 2.3 Contribution 5 2.4 Research Approach 5 Chapter 3: Background 6 3.1 Microelectromechanical System History 6 3.2 Microelectromechanical System Fabrication 7 3.2.1 Bulk Micromachining 7 3.2.2 Surface Micromachining 8 3.2.3 LIGA (Lithographie, Galvanoformung, Abformung) 8 3.2.4 New Fabrication Methods 9 3.3 3D Microelectromechanical Systems 10 3.4 Ortho-Planar Mechanisms 11 3.5 Spherical Mechanisms 11 3.5.1 Spherical Trigonometry 12 3.6 Compliant Mechanisms 13 3.6.1 Small-Length Flexural Pivots 14 3.7 The Pseudo-Rigid-Body Model 15 3.8 Virtual Work Analysis 16 3.8.1 General Form of a Torsional Spring 16 3.8.2 Virtual Work 17 Chapter 4: Mathematical Model of the Bistable Spherical Compliant Mechanis m 18 4.1 Spherical Geometry 18 4.1.1 Spherical Closed Form Equations 19 4.2 The Twisting and Bending Model 19 4.3 Virtual Work 21 4.3.1 First Equation of Virtual Work 21 4.3.2 Second Equation of Virtual Work 24

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ii Chapter 5: The Finite Element Model 26 Chapter 6: Results 29 6.1 The Pure Bending Model 29 6.2 The Variable Parameter: The Length of Compliant Joint 2 30 6.3 The Twisting and Bending Model 30 6.4 Macro Scale Models 32 6.5 Finite Element Analysis: Beam Element Model 34 6.6 Static Jumps 36 6.7 Bistability 37 Chapter 7: Conclusion 38 References 39 Appendices 41 Appendix A: ANSYS Batch Files 42 Appendix B: MATLAB M-Files 61

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iii List of Tables Table 1 Model Parameters 30

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iv List of Figures Figure 1. A scanning electron microscope image of a micro bistable spherica l compliant mechanism 2 Figure 2. The bistable spherical compliant mechanism design and nomenclature 3 Figure 3. (a) The bistable spherical compliant mechanism configuration af ter an applied input, (b) A close-up of the resultant twisting in compliant joint 2 due to an applied input 4 Figure 4. Great circles, great arcs, dihedral angles, and a spherical tr iangle shown on a sphere 12 Figure 5. A small-length flexural pivot 14 Figure 6. A simplified diagram of the bistable spherical compliant mechani sm in the form of a partially compliant four-bar mechanism 15 Figure 7. The bistable spherical compliant mechanism pseudo-rigid-body model 16 Figure 8. Spherical diagram and nomenclature for the bistable spherical compliant mechanism 18 Figure 9. Pseudo-Rigid-Body model of link 2: Part (a) front view, Part (b) top view 20 Figure 10. Plot of the nodal configuration and constraints used to obtain finite element analysis results 27 Figure 11. For Length 5, plots of input moment, M 2 and the input link rotation, 2 for the pure bending model and finite element model 29 Figure 12. Plot of twisting and bending model input moment, M 2 and the input link rotation 2 31 Figure 13. A macro scale model of the bistable spherical compliant mechanism undeflected 33

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v Figure 14. A macro scale model of the bistable spherical compliant mechanism showing the twist in compliant joint 2 33 Figure 15. Compliant joint 2 in a twisted configuration 34 Figure 16. Plot of finite element model input moment, M 2 and the input link rotation, 2 35 Figure 17. Plot of input moment, M 2 and the static jumps that occur during counter-clockwise and clockwise rotation of the input link, 2 36 Figure 18. Plot of bistable positions of the bistable spherical compliant mechanism 37

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vi Abstract The bistable spherical compliant mechanism (BSCM) is a novel device capable of large, repeatable, out-of-plane motion, characteristics that are som ewhat difficult to achieve with surface micromachined microelectromechanical syst ems. An improved pseudo-rigid-body model (PRBM) to predict the behavior of the BSCM is presented. The new model was used to analyze seven different versions of the devic e, each with a different compliant joint length. The new model, which adds torsion, is c ompared with a finite element analysis (FEA) beam model. The new model more cl osely approximates the results yielded by FEA than previous models used to analyze t he BSCM. Future work is needed to quantify stress-stiffening interactions between bendin g and torsion. Both FEA and the current models show that increasing the length of the compliant segment decreases the amount of force required to actuate the device.

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1 Chapter 1 Introduction Integrated circuit technology has allowed for the fabrication of i ncreasingly complex micromechanisms, the majority of micromechanisms being f ound in a variety of sensors and optical switches. Although these sensors are used in a wi de range of applications, they do not exhibit large mechanical motion. The displace ments demonstrated by these micro mechanisms are, in the case of a sensor, most often used to measure a change through piezoresistive, thermal, or capacitive tra nsduction methods (1), or in the case of a micro-mirror, alter the mirror's angl e. A device such as the micromirror performs the largest displacement, but it is still a r elatively small fraction of the size of the device. The development of large mechanical displaceme nt is an area vital to expanding the functionality of micromechanisms. The bistable spheric al compliant mechanism (BSCM), shown in Figure 1, exhibits large, repeatable, bist able, out-of-plane motion, when given a planar input, which provides a gateway to enhanced micromechanism functionality. Compliance in MEMS allows designs t o be greatly simplified and averts the complications associated with micro-hin ges. Bistability allows precise and accurate movement between two stable positions. Although t he BSCM concept has previously been reported in a thesis by Choueifati (2), h ere we provide an alternative analytical model and an analysis of seven different configurations each with a different compliant joint length.

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2 Figure 1. A scanning electron microscope image of a micro bistable sphe rical compliant mechanism 1.1 The Bistable Spherical Compliant Mechanism The BSCM was fabricated using the Multi-User MEMS Process ( MUMPS), a three-layer surface-micromachining method for polysilicon. Spheric al geometry was used in the design of the mechanism. The BSCM has three basic components: two sliders and a compliant spherical four-bar mechanism, with links r 1 r 2 r 3 and r 4 as shown in Figure 2. The ground link is r 1 the input link r 2 r 3 is the coupler link, and r 4 the follower link. Links r 2 and r 4 are joined to the substrate by a staple hinge that allows 180 o rotation. Link r 3 is connected to r 2 and r 4 by compliant joints, nominally small-length flexural pivots (SLFPs) which are discussed in section 3.6.1. The two sliders act a s mechanical actuators and are connected to the input link, r 2 by staple hinges (2). An analytical model in the form of a pseudo-rigid-body model (PRBM) is used to analyze the me chanism and compare with results obtained from a finite element analysis (FEA).

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3 Figure 2. The bistable spherical compliant mechanism design and nomenclature

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4 Chapter 2 Outline 2.1 Objective The objective of this research is to improve the PRBM of the BSCM by including the torsion in joint 2 to a previous PRBM and comparing it to results obt ained through FEA. The primary factor affecting the torsion of the device i s assumed to be the length of compliant joint 2, thus seven versions of the device are analyzed, each with a different compliant joint length, as shown in Figure 2. The torsion in compliant joi nt 2 can be seen in Figure 3. (a) (b) Figure 3. (a) The bistable spherical compliant mechanism configur ation after an applied input, (b) A close-up of the resultant twisting in compl iant joint 2 due to an applied input 2.2 Motivation The addition of new micromechanisms capable of large out-of-plane mot ion is crucial to the development of miniaturization technology. The lack of new designs is primarily due to fabrication and assembly limitations on the micr oscale, and the lack of

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5 design processes capable of predicting input-output, or force-deflect ion relationships, necessary to design micromechanisms to meet predetermined criteria in order to complete specific objectives. Spherical compliant mechanisms alleviate ma ny of the complications associated with microfabrication, thus accurate PRBMs greatly benefit the advancement microsystem design. 2.3 Contribution An improved PRBM for the BSCM is presented and compared to FEA re sults for seven different versions of the BSCM. The examination of the differ ent versions of the device allowed for quantitative similarities to be observed indicat ing that the new PRBM captures the main characteristics of the input-output relationships. 2.4 Research Approach A review of the various fields of knowledge applicable to the BSCM ’s fabrication, design, and analysis is presented to provide a background and reinforce the contribution of this work. The mathematical model and theory of the BSC M is then expressed, followed by the approach taken toward a finite element a nalysis. Finally, results and conclusions are presented.

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6 Chapter 3 Background The BSCM is designed using a multitude of methods from various fiel ds of knowledge. An overview of the associated fields is presented, specifica lly: MEMS history, integrated circuit (IC) fabrication methods, 3D MEMS, ort ho-planar mechanisms, spherical mechanisms, compliant mechanisms, pseudo-rigi d-body models, and the method of virtual work. 3.1 Microelectromechanical System History The possibilities and difficulties of MEMS were realized during the 1960s, and the first silicon based micromechanical structures were devel oped as early as 1965 (3). During the 1970s, numerous fabrication advancements were made that spur red the evolution of some of the most prevalent MEMS devices produced today. One of the earliest MEMS to become prevalent in industry was the pressure sensor, created using IC compatible technologies such as doping, thin film materials, and bondi ng (4). The importance of silicon as a mechanical material and its numerous f abrication methods were outlined in the 1980s (5)(6). A large drive from the automotive i ndustry, based on a need for batch fabrication at low cost and reduced size, helped bolster the development of MEMS. The automotive industry benefits from the use of an array of micro-sensors including, accelerometers, mass/air flow sensors, load/force sensor s, torque, position, temperature, light, and oxygen sensors (1).

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7 Other applications of MEMS include structural health monitoring of critical aircraft components, bio-medical applications, micro-actuators, tele communications, and IT peripherals such as mobile phones, PDA’s, laptops, and game consoles (7)(8)(9). Advancements in MEMS to these areas will continue to advance MEMS technology. More recent and future applications of MEMS, include micro mirrors, microphones, ink jet printing heads, and micro-spray nozzles (8). 3.2 Microelectromechanical System Fabrication The most common methods for MEMS fabrication are bulk micromachining, surface micromachining, and LIGA (Lithographie Galvanoformung Abf ormung). Newer methods of MEMS fabrication include deep reactive ion etching (DR IE), extreme ultraviolet lithography (EUVL), charged particle beam lithograph y (E-Beam Lithography, EBL), ion beam lithography (I-Beam Lithography, IBL), and electrochemical fabrication (EFAB) (1). MEMS materials ar e limited to silicon, its compounds, glasses, and some metals, but it is gradually extending t o more metals, ceramics, and polymers (10). Although the BSCM was fabricated us ing surface micromachining, each of the aforementioned fabrication methods were r esearched in order to asses if the design of the BSCM could be improved and to see if any other devices have been produced that are similar to the BSCM. 3.2.1 Bulk Micromachining Bulk micromachining is a relatively fast and inexpensive method of fabricating MEMS due to its single substrate layer of single crystal m aterial, most commonly silicon.

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8 The silicon substrate is patterned using photolithography. A chemica l etchant, such as potassium hydroxide, is used to remove selected areas. After etchi ng is complete, the desired structure remains (1). Bulk micromachining cannot produce a des ign as complex as the BSCM. 3.2.2 Surface Micromachining Surface micromachining differs from batch micromachining in that multiple layers are utilized, and etched, on top of an untouched substrate. This t echnology stems from IC’s where dozens of layers is not uncommon. However, micromech anisms generally do not exceed six layers (1). The separate layers used in surface micromachining are term ed structural and sacrificial layers. Structural layers usually consist of a thin film of polysilicon, sacrificial layers, silicon dioxide. These thin films are generally one to two micrometers thick. The polysilicon is deposited using low pressure chemical vapor deposition a nd the silicon dioxide layers are grown by thermal oxidation. Each consecutive la yer is patterned using photolithography and etching. Finally, the sacrificial layers ar e dissolved using hydrofluoric acid and the mechanism is said to have been releas ed (11). This layered process allows the links of the BSCM to be fabricated free of the substrate, attached only by the staple hinges, so that actuation is possible. 3.2.3 LIGA (Lithographie, Galvanoformung, Abformung) LIGA is an acronym for the German words lithographie galvanofor mung abformung, meaning lithography, electroplating, and molding (12). LIGA use s x-ray

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9 radiation produced by a synchrotron to pattern a polymethyl methacr ylate substrate into a mold. The mold is then electroplated with nickel, copper, or nickel iron. A chemical rinse is used to dissolve the mold leaving the metal part free and rea dy for assembly. The metal part may also be used as a mold to form a polymer or ceramic pa rts by injection molding. Advantages to LIGA are high aspect ratios, on the order of 100:1, and a greater selection of materials. The number of facilities able to perform LIGA is limited due to the necessity of a synchrotron, and the assembly of parts is usually required, which can be difficult (1). 3.2.4 New Fabrication Methods There is a gradual shift toward using the term microsystem t echnology, as opposed to MEMS, as miniaturization science begins to expand to include systems that focus on dynamic motion without the need for integrating electrical c ircuitry. IC fabrication methods have thus become more diverse regarding the use of lithography techniques. IC's priorities require throughput, finer geometries, and ba tch processing whereas miniaturization science requires modularity, good depth of foc us, extending the z-direction, incorporating nontraditional materials, replication, and ba tch fabrication is not always a prerequisite (1) Newer fabrication methods such as, Extreme Ultraviolet Lithogr aphy (EUVL) (13), Deep X-Ray Lithography (DXRL) (14), Charged Particle Bea m Lithography (EBeam Lithography, EBL) (15), and Ion Beam Lithography (I-Beam Li thography, IBL) (16), succeed in achieving high aspect ratios, some at multiple an gles to the substrate, with varying fabrication difficulties ranging from simple a nd fast methods to complex

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10 methods that require controlled work environments. Regardless of the im provement, the outcome is still dominantly planar, where additional benefits of an gled designs can be negated by restricted complexity of design. Relatively organic shapes can be produced using a fabrication method termed EFAB (electrochemical fabrica tion). EFAB has the ability to produce true 3-D structures due to its ability to use a significant number of layers, several dozen, compared to say Sandia's SUMMIT V progra m which comprises of five layers. Advances in MEMS fabrication have produced finer line widths, higher features, high aspect ratios, and the introduction of new materials. These new m aterials, such as SiC, allow MEMS to perform in environments with extreme temperat ure, vibration, and harsh chemical media. Polymers are becoming more prevalent due to their increased fracture strength, low Young's modulus, high elongation at break, and their compatibility in biological and chemical applications (17). Complex ortho-planar mec hanisms continue to elude fabrication due to the limited capabilities of micromac hining, microassembly, and layered fabrication methods imposing planar designs of MEMS (10) Surface micromachining is still the optimal choice for fabricating the BSCM. 3.3 3D Microelectromechanical Systems Recent advancements to MEMS fabrication have focused on improving cur rent planar methods to produce finer line widths and increased design heights but the complexity of designs remains near the same (16)(17). Organic sha pes may be fabricated using a high number of layers through methods such as EFAB, or microstereolithography, and the assembly of parts fabricated with robust materials

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11 through LIGA fabrication show promise for the future. Benefits of m icroassembly include combining parts that use different or incompatible material s or manufacturing processes and the ability to increase the number of layers of a de vice to form true 3D mechanisms. A review of micro-scale assembly technologies has b een carried out by Cohn et al. (1998) and Probst et al. (2009) (9). EFAB, although able to cr eate organic shapes, is still limited to planar design limitations, and microa ssembly is not economically viable due to time consuming assembly techniques. 3.4 Ortho-Planar Mechanisms Orthogonally-Planar or ortho-planar mechanisms allow motion in one plane to be transferred in a direction perpendicular to that plane. Ortho-Planar designs are therefore able to take advantage of the planar nature of IC fabrication methods and can produce motion in the z direction, greatly increasing the functionality of MEMS devi ces. 3.5 Spherical Mechanisms A spherical mechanism is one in which all axes of rotation of its revolute joints intersect at a single point that defines the origin of the sphere. Circles drawn on the surface of the sphere and in a plane that intersect the center of the sphere are termed great circles. The intersection of two great circles on a sphere def ines a dihedral angle. The intersection of three great circles defines the sides of a spherical triangle, and the sides of the spherical triangle are termed great arcs. Each great arc has a corresponding dihedral angle that is opposite to it (18). Great circles, great arcs, di hedral angles and a spherical triangle are shown in Figure 4.

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12 Figure 4. Great circles, great arcs, dihedral angles, and a spherical tri angle shown on a sphere 3.5.1 Spherical Trigonometry Much like planar geometry, laws of sines and cosines exist that are specific to spherical triangles (19). The spherical law of cosines is lat er used to derive the closed form equations of the BSCM. The spherical law of cosines can be appl ied to a spherical triangle using either three arcs and one dihedral angle or three dihedral angles and one arc. In the following equations, great arcs are represented b y A B and and dihedral angles are represented by and as shown in Figure 4.

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13 Three arcs and one dihedral angle. cos sin sin cos cos cos E B E B A n (1) Three dihedral angles and one arc: A cos sin sin cos cos cos n (2) 3.6 Compliant Mechanisms A Compliant mechanism transfers, or transforms motion, force, or ene rgy using deflection of flexible members rather than moveable joints (21). The advantage of compliant mechanisms at the micro level are mainly due to defle ction as opposed to constraints due to pin joints (10)(21). Compliant mechanisms reduce the total number of parts, reduce friction and wear, require less space and are less complex, have less clearance due to pin joints resulting in higher precision, require no a ssembly, integrate energy storage elements (springs) with other components, and can be f abricated in a plane, all of which are desirable in MEMS (21)(23). These benefits overcome mechanical problems such as: binding, clearances which are large compared to de vice sizes, friction, and wear (20). Compliant mechanisms are often monolithic adding to its desirability for batch fabrication of assembly free components, or entire devices, and complex motions are still achievable even with the constraints of micromachining (10). A disadvantage of compliant MEMS is that performance is highly dependent on material properties and the selection of materials is limite d to the few materials available for MEMS fabrication. Material properties do not always behave as predicted due to model sizing being of the same order as material grain size. E ven so, compliant mechanisms can be very robust at the micro level and often last long er than do

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14 components that use pin joints or other elements that induce wear. Compli ant mechanisms bridge the gap between planar fabrication and large out of plane motion by simplifying designs while achieving repeatable out of plane mot ion. There will continue to be a requirement for compliant mechanisms due to their precisi on, repeatability, and ability to perform bistability. 3.6.1 Small-Length Flexural Pivots A small-length flexural pivot is a type of compliant joint tha t obtains its characteristic motion due to isolated bending in the short segment of the beam, as shown in Figure 5. The wider segment of the beam remains essentia lly straight due to its relatively high rigidity compared to the low rigidity of the narr ower segment. The BSCM's compliant segments allow for the large deflections ex hibited by the mechanism. Additional benefits of compliance in microsystems include: simplif ication of design, reduced part count, reduced wear, and lower production cost (21). Figure 5. A small-length flexural pivot. Adapted from Howell

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15 The replacement of two rigid-body pin-joints with small-length fl exural pivots can be seen in the simplified diagram of the BSCM as shown i n Figure 6. Although compliant joints 2 and 3 are fully compliant, the mechanism as a wh ole is termed 'partially compliant' due the combination of rigid-body joints and compl iant joints. The partially compliant four-bar represented in Figure 6 is used as the basis for the pseudorigid-body model discussed next. Figure 6. A simplified diagram of the bistable spherical compliant mechanism in the form of a partially compliant four-bar mechanism 3.7 The Pseudo-Rigid-Body Model Pseudo-Rigid-Body models can be used to replace flexible membe rs with rigidbody components that have equivalent force-deflection characteristics. Rigid-Body mechanics can then be used to analyze the compliant system (21). In the BSCM, smalllength flexural pivots are replaced with pin joints and torsional spr ings, as shown in Figure 7, where k 1 k 2 and k 3 are the small-deflection stiffness constants.

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16 Figure 7. The bistable spherical compliant mechanism pseudo-ri gid-body model The pseudo-rigid-body model allows for the method of virtual work to be applied to the mechanism. 3.8 Virtual Work Analysis The method of virtual work is used to derive the coupling of rigid links due to the presence of the torsional springs in the pseudo-rigid-body model. The virtual work is found by taking the derivative of the strain energy of the torsi onal springs and multiplying it by their respective virtual displacements (21). 3.8.1 General Form of a Torsional Spring The general equation for torsional spring strain energy can be written as r o d m V o o k m (3) where m is the moment.

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17 After integration, the torsional spring strain energy, V becomes 2 2 1 o k V (4) 3.8.2 Virtual W ork The virtual work is found from the derivative of the torsional spring s train energy with respect to the generalized coordinate to be q dq dV W (5) q dq d m W (6) The virtual work due to a moment on a rigid link, M i and a virtual angular displacement, q i where q i is the generalized coordinate, is i i q M W (7) The virtual work due to a moment at a small-length flexural pivot, T i and a virtual angular displacement, q i where q i is the generalized coordinate is i i q T W (8) The total virtual work of the system is i i i i q T q M W n (9)

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18 Chapter 4 Mathematical Model of the Bistable Spherical Compliant Mechanism 4.1 Spherical Geometry The spherical geometry nomenclature of the BSCM is shown in Figure 8, where r 1 r 2 r 3 and r 4 represent great arcs and and represent the dihedral angles. The arc is used to separate the spherical four-bar into two spherical tr iangles in order to apply spherical trigonometry. Figure 8. Spherical diagram and nomenclature for the bistable spheric al compliant mechanism

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19 4.1.1 Spherical Closed Form Equations The variables and are derived using the background outlined in section 3.5.1. Using the spherical law of cosines, equation (1), and Figure 8 one finds: 2 2 1 2 1 1 cos sin sin cos cos cos r r r r n (10) sin sin cos cos cos cos 2 2 1 1 r r r (11) sin sin cos cos cos cos 3 3 4 1 r r r (12) 4 3 4 3 1 sin sin cos cos cos cos r r r r (13) sin sin cos cos cos cos 4 4 3 1 r r r (14) sin sin cos cos cos cos 1 1 2 1 r r r (15) These equations allow the calculation of the motion of the BSCM for a given input The angle is equal to the input rotation in the pure bending model and in the twisting and bending model as discussed next. 4.2 The Twisting and Bending Model To solve for the torsion of compliant joint 2, it is necessary to know two angles to define the angle of twist. In this case, the primary angle is 2 and the secondary angle, The primary angle, 2 is the exact value of the input link, r 2 as it is rotated off of the substrate; consequently, it is also the angle at the end of complia nt joint 2 that connects to the input link, r 2 The secondary angle, defines the angle of twist in compliant joint 2. It is the angle at the end of compliant joint 2 that connects to the coupler link, r 3

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20 As shown in Figure 9, the angle shares the same axis of rotation as 2 From a kinematic standpoint (neglecting the torsional spring that couples their motion), the motion of the spherical four-bar depends entirely on the rotation of and not at all on the rotation of 2 The method of virtual work is used to derive the coupling due to the presence of the torsional spring in the PRBM. Figure 9. Pseudo-Rigid-Body model of link 2: Part (a) front view, Part (b) top view A system of two equations was necessary to solve for as a function of 2 The first equation of virtual work is derived with 2 as the independent variable and therefore concentrates on the input link, r 2 The second equation of virtual work is derived with as the independent variable and includes the twisting and bending associ ated with compliant joint 2 and the bending of compliant joint 3. The second equation of vi rtual

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21 work captures the deflection of the mechanism that is responsible for the twisting in compliant joint 2. The method of virtual work was also used to solve for the moment of joi nt 1 as a function of 2 In other words, the moment required to actuate the BSCM for a gi ven input. 4.3 Virtual Work 4.3.1 First Equation of Virtual W ork The first equation of virtual work is used to form a correlation betw een the generalized co-ordinates and This correlation allows us to solve for which is necessary to define the twist. The virtual work with the generalized co-ord inate as is 3 3 2 2 1 1 2 2 n n n T T T M W (16) 2 2 3 3 3 2 2 2 2 2 1 1 1 2 0 m m m M (17) The moments associated with the torsion and bending of compliant joint 2, are m 1 and m 2 respectively, and the moment associated with the bending of co mpliant joint 3, m 3 are defined as 1 1 1k m 2 2 2k m 3 3 3k m (18) where k 1 k 2 and k 3 are small-deflection stiffness constants found in Ugural et a l. (22). The value k 1, is the torsional spring constant, and k 2 and k 3 are the bending spring constants. Each spring constant is dependent on the material and ge ometric properties of the compliant joints of the BSCM. J is the polar moment of inertia G is the shear modulus of elasticity L is the length of the compliant segment, E is Young's modulus I is

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22 the second moment of area v is Poisson's ratio and is a constant dependant on the width and height of the rectangular cross section of compliant joint 2 (22). The torsional small-deflection stiffness constant is defined as 2 1 L JG k (19 ) where the product JG is the torsional rigidity (22) and the length L 2 is the length of compliant joint 2. 3 ab J T (20) 3 wh J T 2125 .0 T (21) n 1 2 E G MPa E 3 10 169 35.0 (22) The bending spring constants are defined as 2 2 2 L EI k (23) 2 2 3 2 L wh E k (24) 3 3 3 L EI k (25) 3 3 3 3 L wh E k (26) The angular displacements of the torsion of compli ant joint 2, the bending of compliant joint 2, and the bending of compliant joi nt 3 are defined by and respectively. The angle is the difference between the primary and secondary angles of compliant joint 2, which defines the angle of tw ist, as seen in Figure 9. 2 1 (27)

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23 The angle is the angle between links r 2 and r 3 as defined in Figure 8, and the angle" is the angle between links r 3 and r 4 also defined in Figure 8. n o o 2 o 3 (28) The partial derivatives of the angular displacemen ts with respect to the generalized co-ordinate are termed "kinematic coeff icients" (sensitivities) and are determined in eq. (24 & 25). They are based on the assumption that and are kinematically independent but statically coupled du e to the torsional spring. In this way, the kinematic coefficient of torsion in compliant j oint 2 becomes 1 2 1 (29 ) and the kinematic coefficient of bending in complia nt joints 2 and 3 both become 0, as they do not depend on the generalized co-ordinate 0 2 2 0 2 3 (30) Equation (30) simplifies equations (16 & 17) to 1 2 0 T M n (31) Equation (31) can now be substituted into the seco nd equation of virtual work to solve for

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24 4.3.2 Second Equation of Virtual W ork The second equation of virtual work is based on th e generalized co-ordinate and takes into consideration both the torsion and b ending of joint 2 and the bending of joint 3. The spring stiffness', k 1 k 2 and k 3 and the angular displacements and are the same as in the first equation of virtual wo rk. 3 3 2 2 1 1 2 2 n n n T T T M W (32) 3 3 3 2 2 2 1 1 1 2 2 0 m m m M (33) The first kinematic coefficient is zero due to 's independence of 0 2 (34) The torsion and bending kinematic coefficients are based on the partial derivatives of equations (27 & 28) with respect to where the angles and are found in equations (10-12) and are dependent on in equation (9). For simplicity, c ( x ) =cos ( x ), s ( x ) =sin ( x ), s 2 ( x ) =sin 2 ( x ), etc. 1 1 (35) n 2 (36) c r c c r c s r c s s s r s 1 2 2 2 2 3 1 n (37) c r c c r c s r c s s r s s r s r s 4 2 3 2 3 3 3 2 1 n (38) s r s r s s r s r s 4 3 2 1 (39 )

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25 From equation (33) and substituting equations (3439) we find: n n 3 3 2 2 1 0 T T T (40) n n 3 3 2 2 2 1 0 T T k (41) n 3 3 2 2 1 2 1 T T k ( 42 ) From equation (33) and substituting equations (31 & 34-39) we find: n n 3 3 2 2 1 0 T T T (43) n n 3 3 2 2 2 0 T T M (44) n 3 3 2 2 2 T T M (45 ) The moment required to actuate the device can now be plotted against the input, 2

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26 Chapter 5 The Finite Element Model The finite element model was constructed in ANSYS using the BEAM4 beam element model, a 3D elastic beam. The large deflection of the BSCM required the use of the non-linear iterative solver. The BEAM4 model all ows six degrees of freedom, three in translation, and three in rotation. The beam is con strained to allow rotation about the Y axis at two points: one at the staple hinge of joint 1, and one also on link, r 2 but before compliant joint 2. The staple hinge at joint 4 is also constrained to only allow rotation about the x and y axis in Figure 10. The staple hing es can be viewed in Figure 2 and the constraints can be viewed in Figure 10.

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27 Figure 10. Plot of the nodal configuration and constraints used to obtain f inite element analysis results Obtaining the FEA results took a great deal of time d ue to singularities encountered while rotating the model from its initial position, as shown in Figure 10, through -180 o The sign of the rotation of the model is based on the right hand rule applied to Figure 10. Each time a singularity was enco untered, the program failed to converge and no data was stored. Multiple singulariti es were encountered for each model as a result of the BSCM storing potential energy and t hen releasing that potential energy, causing the mechanism to flip into a new orientation fo r a single value of input. This phenomenon is examined in chapter 6. Each time a singula rity was encountered, the program needed to be switched from displacement to force mode, or vice-versa. A counter was placed within the program in order to gai n an understanding of the amount of

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28 steps completed before divergence occurred and the pro gram ended. Although the counter provided an approximate location of the dive rgence, it did not discern the step size required to successfully reach the end of one mode, to continue the program in the alternate mode. If the step sizes were large, the progr am would end and it would not be known how far to turn back. If the step sizes were too small, the program could be ran successfully, but it would not be known how far the pr ogram could go before a singularity was reached. Thus, the results were achieve d through trial and error and can be viewed in Figure 16.

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29 Chapter 6 Results 6.1 The Pure Bending Model Choueifati (2) described the pure bending model in 2 007, in which the bending of compliant joints 2 and 3 were analyzed, but not the torsion. The results of Choueifati’s model is provided in Figure 11 to compare to new resul ts achieved by the twisting and bending model discussed in the next section. Figure 11. For Length 5, plots of input moment, M 2 and the input link rotation, 2 for the pure bending model and finite element model The model yields a smooth curve, some initial similarities t o the FEA model, and captures the change in direction of the moment. It doe s not incorporate the twisting in compliant joint 2.

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30 6.2 The Variable Parameter: The Length of Compliant Joint 2 The twisting and bending model, discussed next, differs based on the length of compliant joint 2. Seven different lengths of compli ant joint 2 are studied; however, the width and height of compliant joint 2 remains constant. The geometry of the rigid members remains constant, as does the geometry of compliant joint 3, as shown in Table 1. Table 1. Model Parameters Width ( m) Height ( m) Length ( m) Rigid Members 25 2 ---------Joint 3 5 2 22.17 Joint 2 Length 1 3.5 2 30 Length 2 3.5 2 40 Length 3 3.5 2 50 Length 4 3.5 2 60 Length 5 3.5 2 80 Length 6 3.5 2 100 Length 7 3.5 2 120 6.3 The Twisting and Bending Model The twisting and bending model incorporates the twisti ng that occurs in compliant joint 2 to the PRBM. The results of the new model are shown in Figure 12. It is immediately apparent that the results of the new model achieve a significant improvement over the results of the previous pure bendin g model of Figure 11. The twisting and bending model captures the highly non-lin ear characteristics of the BSCM that are also seen in the FEA analysis as shown in Figur e 16. The highly non-linear behavior is a result of the compliance and stiffness of t he BSCM, where the stiffness is indicated by the slope of the line. Although the cor rect stiffness trend is captured, the trend in the maximum moment is only partially correct.

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31 The trend in the moment is easiest described by looking at values above and below the value of zero moment. Below the value of ze ro moment, in the negative values, the twisting and bending model shows a slight increase in the maximum moment to actuate the device as the length of compliant joint 2 increases. FEA however, shows the opposite trend, a decrease in the maximum moment, as shown i n Figure 16. The FEA model shows the correct trend and although the twisting and bending model does not, future work will improve the trend. Above values of z ero moment, the positive values, the correct trend does exist in the twisting and bend ing model. Both the twisting and bending, and FEA, models indicate an increasing maximum moment as the length of compliant joint 2 increases. Figure 12. Plot of twisting and bending model input moment, M 2 and the input link rotation 2 The highly non-linear behavior of the BSCM, as shown in Figure 12, is in part due to the mechanism having multiple moment values for a si ngle input rotation value.

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32 This is a result of the compliant segments storing potentia l energy, and at a certain point, releasing it back into the mechanism. The rotation of t he input link, r 2 produces a displacement of the follower link, r 4 that is controlled by the rotation of the input li nk and the potential energy in the system. As the input link, r 2 is rotated, 2 increases, as does # but at a slower rate due to storage of potential en ergy in the compliant joints. The input link rotation, 2, is able to increase to a maximum value termed by Young et al. (24) as 'the load threshold', at which point the stored pot ential energy is converted to kinetic energy and the mechanism moves spontaneously, or jumps, from one statically stable position to a different one. The tendency for the mec hanism to jump is the cause of the multi-valued plots of Figure 12 and Figure 16, and is discussed further in section 6.6. 6.4 Macro Scale Models It is observed in Figure 12, that as the length of th e SLFP of compliant joint 2 increases, the moment required to actuate the device i ncreases. This is not the intuitive relationship one would deduce from adding increased fl exibility to the mechanism. The results of the FEA analysis, Figure 16, show that increa sing the length of compliant joint 2 decreases the moment required to actuate the device. This is what one would expect due to the reduced torsional spring constant associated with longer beam lengths where low torsion and high rotation would occur before the mech anism achieved large displacements. Macro scale models supported this theory by exhibiting the characteristics portrayed by the FEA results. An example of the macro sc ale models can be seen in Figure 13, Figure 14, and Figure 15.

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33 Figure 13. A macro scale model of the bistable spherical compliant me chanism undeflected Figure 14. A macro scale model of the bistable spherical compliant me chanism showing the twist in compliant joint 2

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34 Figure 15. Compliant joint 2 in a twisted configuration 6.5 Finite Element Analysis: Beam Element Model The finite element analysis results in Figure 16 indica te qualitative similarities compared to the new analytical model. Although quali tative similarities are observed, it is noted that the moment is greater than in the analyti cal model, suggesting that our current analytical model underestimates the stiffness of t he mechanism. Thus, the bending and torsional stiffness' given in equation (19 24, 26), do not account for stressstiffening interactions that occur when a bent beam is twisted or when a twisted beam is bent. Building the FEA model required switching between l oad control and displacement control as singularities were encountered. As each singularity was reached, the final load step was recorded, in order to begin t he program again starting from the last load step but in the new control mode.

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35 Figure 16. Plot of finite element model input moment, M 2 and the input link rotation, 2

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36 6.6 Static Jumps The plots in Figure 12, and Figure 16, predict some un usual behaviors. As a moment is applied to the input link, r 2, rotation occurs until the load threshold of 2 is reached, at which point multiple values of M 2 are seen for the same rotation. In order to continue motion, the plots suggest that the moment must be released and then applied in the opposite direction, to allow a decrease in rotati on before resuming the desired counter-clockwise rotation. This suggested release of mome nt and backwards motion is not characteristic of four-bar motion, or compliant me chanisms, and does not exist in physical models. Rather, as a moment is applied to the inp ut link, r 2 rotation occurs to its initial maximum value, and its corresponding initial mome nt, and then jumps to a second value of moment before continuing to the final point of rotation at -180 o This phenomenon was termed a static jump by Young et al. (24) and is shown graphically in Figure 17. Figure 17. Plot of input moment, M 2 and the static jumps that occur during counter-clockwise and clockwise rotation of the input link, 2

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37 6.7 Bistability The deflection of compliant members incorporates both mo tion control and energy storage leading to the possibility of accurate and precise bistable mechanisms without the use of joints and springs. Bistability is a h ighly desired characteristic in micromechanisms because power is not necessary to hold the me chanism in either of its stable positions (25). The BSCM has two stable positions at 0 o and 180 o as shown in Figure 18. Figure 18. Plot of bistable positions of the bistable spherical comp liant mechanism

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38 Chapter 7 Conclusion The bistable compliant spherical mechanism adds a new mec hanism to the very limited field of ortho-planar micromechanisms. The additi on of twist to the analytical model shows a marked improvement over the previous model an d allowed for the quantification of the dynamic characteristics unique t o the bistable spherical compliant mechanism. The analysis of seven different lengths of comp liant joint 2 allowed for the primary factor affecting the torsion of compliant joi nt 2 to be examined. Quantitative similarities are shown indicating that the model capture s the main characteristics that define the input-output relationships but the model sti ll underestimates the overall rigidity. Future work will determine the appropriate pseudo-rigid-body model for mechanisms with bending and torsion interactions.

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39 References (1) Madou, M. (2002), Fundamentals of microfabrication Boca Raton, NY: CRC Press. (2) Choueifati, J. G. (2007). Design and modeling of a bi stable spherical compliant mechanism, M.S. Thesis. University of South Florida (3) Frazier, B. Warrington, R.O. Friedrich, C. (1995). The miniaturization technologies: past, present, and future. IEEE Transactions on Industrial Electronics 42 no. 5, pp 423-430. (4) Smith, J.H. Eaton, W.P. (1997). Micromachined pressure sensors: review and recent developments. Office of scientific and technica l information (5) Lee, A. P. (1996), Microelectromechanical systems (MEMS ) for engineers, Lawrence Livermore National Laboratory, Livermore, C A. (6) Fujita, H. (1997). A decade of MEMS and its future. Institute of Industrial Science, University of Tokyo. (7) Polla, D. (1997), MEMS for integrated diagnostic app lications, American Society of Mechanical Engineers Tribology Division, vol 7, pp. 19-24. (8) Urban, A., Laermer, F., (2008), Bosh process-DRIE succ ess story, new applications and products, Proceedings of the Materi al Research Society, Jan. (9) Tamadazte, B. Marchand, E. Dembele, S. Li Fort-Piat, N. (2010). CAD modelbased Tracking and 3D visual based control for MEMS mi croassembly, international journal of robotics research, vol. 29, no. 11, pp 1416-34. (10) Ananthasuresh, G. K. Howell, L. L. Mechanical design o f compliant microsystems-a perspective and prospects, 2005, Journal of Mechanical Design vol. 127, no. 4, pp 736-738. (11) Sniegowski, J.J. (1996). Multi-Level polysilicon surfa ce micromachining technology: applications and issues. international mech anical engineering congress and exhibition, Atlanta, GA, Nov 17-22. (12) Bonivert, W.D. et al. (1999). LIGA micromachining: in frastructure establishment. Office of scientific and technical information

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40 (13) Anderson, C.N. (2009). Extreme ultraviolet lithograph y: a few more pieces of the puzzle, Lawrence Berkeley National Laboratory; Oak Ridge, Tenn (14) Green, G.K. (1977). Synchrotron radiation: its chara cteristics and applications. Brookhaven National Laboratory ; Oak Ridge, Tenn (15) Liu, J. Vaclav, V. (2007). Synchrotron facilities an d free electron lasers. United States. Dept. of Energy ; Oak Ridge, Tenn (16) Lindroos, V. Markku, T, Lehto, A. Motooka, T. (2010) Handbook of Silicon Based MEMS Materials and Technologies William Andrew Publishers, Oxford. UK. (17) Adams, T.M. Layton, R.A. (2010). Introductory MEMS: fabrication and applications Springer. New York. (18) Henderson, D. W. (2001). Experiencing geometry in euclidian, spherical, and hyperbolic spaces. Upper Saddle River, NJ: Prentice Hall. (19) Spiegel, M.R. Liu, J. (1999). Schaum’s outlines: mathematical handbook of formulas and tables McGraw-Hill, Washington, D.C (20) Kota, S. Ananthasuresh, G.K. Crary, S. Wise, K.D. (199 4). Design and fabrication of microelectromechanical systems. Journal of Mechanical Design 116, pp 10811088. (21) Howell, L. (2001). Compliant mechanisms New York, NY: John Wiley & Sons. (22) Ugural, C. A., & Fenster, S. K. (2003). Advanced strength and applied elasticity Hamilton in Casleton, NY: Pearson Education. (23) Howell, L. Midha, A. (1994). A method for the design of compliant mechanisms with small-length flexural pivots. Transactions of the ASME 116, pp 280-290. (24) Young, S.O., Kota, S. (2009). Synthesis of multistable equilibrium compliant mechanisms using combinations of bistable mechanisms. ASME. Journal of Mechanical Design. Feb. Vol. 131. (25) Opdahl, P.G. Jensen, B.D. Howell, L. (1998). An inve stigation into compliant bistable mechanisms. Design Engineering Technical Conferences. Atlanta: ASME, pp 1-10.

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41 Appendices

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42 Appendix A: ANSYS Batch Files The following code was originally written and docume nted in a thesis by Choueifati, and has since been amended as documented i n this thesis. Choueifati wrote code for seven different models, but only model 7 was f unctional. The following ANSYS batch files provide a working code for the seven di fferent models. A.1 Batch File 1 !************************************ /CONFIG,NRES,1000000 !/CWD,'C:\Documents and Settings\aleon2\Desktop\Work' !************************************ !**************************************** !******* Set Up Model Variables ********* !**************************************** /title,3D Beam Non-linear Deflection /PREP7 R=315 length in micrometers PI=acos(-1.) h1=2 b1=25 b2=5 h2=2 b3=27.6 h3=2 b4=3.5 h4=2 !*********** Area properties ************** A1 = h1*b1

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43 Appendix A (Continued) Iy1= 1/12*b1*h1*h1*h1 Iz1= 1/12*h1*b1*b1*b1 E1= 169E3 Young's modulus in MPa, Force will be mic ro Newtons !************************************ A2= h2*b2 Iz2= 1/12*h2*b2*b2*b2 Iy2= 1/12*b2*h2*h2*h2 E2= 169e3 !************************************ A3= h3*b3 Iz3= 1/12*h3*b3*b3*b3 Iy3= 1/12*b3*h3*h3*h3 E3= 169e3 !************************************ A4= h4*b4 Iz4= 1/12*h4*b4*b4*b4 Iy4= 1/12*b4*h4*h4*h4 E4= 169e3 !************************************ !********** Declare an element type: Beam 4 (3D Elasti c) ********* ET,1,BEAM4 KEYOPT,1,2,1 KEYOPT,1,6,1 !********** Set Real Constants and Material Propertie s ********* R,1,A1,Iy1,Iz1,h1,b1, !******Check on t he assumptions being made ******

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44 Appendix A (Continued) R,2,A2,Iy2,Iz2,h2,b2, R,3,A3,Iy3,Iz3,h3,b3, R,4,A4,Iy4,Iz4,h4,b4, MPTEMP,1,0 MPDATA,EX,1,,E1 MPDATA,PRXY,1,,0.35 Material proper ties for material 1 and 2 MPTEMP,1,0 MPDATA,EX,2,,E2 MPDATA,PRXY,2,,0.35 MPTEMP,1,0 MPDATA,EX,3,,E3 MPDATA,PRXY,3,,0.35 MPTEMP,1,0 MPDATA,EX,4,,E4 MPDATA,PRXY,4,,0.35 !************************************ !********** Create Keypoints 1 throug 7: K(Point #, X-Coord, Y-Coord, Z-Coord) ********* K,1,0,0,0, K,2,0,-220,0, K,3,0,-15,0, K,4,0,15,0, K,5,R*cos(PI/180*72),R*SIN(PI/180*72),0, K,6,R*cos(PI/180*33),R*sin(PI/180*33),0, K,7,R*cos(PI/180*29),R*sin(PI/180*29),0, K,8,R*cos(PI/180*44),-R*sin(PI/180*44),0, !********* Create Beam using Lines and Arcs and divide into segments ********* LSTR, 2, 3 LSTR, 4, 5

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45 Appendix A (Continued) LSTR, 3, 4 Draws line s connecting keypoints 1 through 7 LSTR, 6, 7 LARC,5,6,1,R, Defines a circular arc LARC,7,8,1,R, LESIZE,ALL,,,15 Specifies t he divisions and spacing ratio on unmeshed lines, *****Try making 32 smaller !*********** MESH *********** real,3 Use real constant set 3 type,1 Use element type 1 mat,3 use material property set 3 LMESH,1,2 mesh lines 1-2 real,2 Use real constant set 2 type,1 Use element type 1 mat,2 use material property set 2 LMESH,4 mesh line 4 real,4 Use real constant set 4 LMESH,3 mesh line 3 real,1 Use real constant set 1 type,1 Use element type 1 mat,1 use material property set 1 LMESH,5,6 mesh line 5-6 !******* Get Node Numbers at chosen keypoints ******* ksel,s,kp,,2 nslk,s *get,nkp2,node,0,num,max Retrieves a valu e and stores it as a scalar parameter or part of an array parameter*********** nsel,all ksel,all ksel,s,kp,,3

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46 Appendix A (Continued) nslk,s *get,nkp3,node,0,num,max nsel,all ksel,all ksel,s,kp,,4 nslk,s *get,nkp4,node,0,num,max nsel,all ksel,all ksel,s,kp,,5 nslk,s *get,nkp5,node,0,num,max nsel,all ksel,all ksel,s,kp,,6 nslk,s *get,nkp6,node,0,num,max nsel,all ksel,all ksel,s,kp,,7 nslk,s *get,nkp7,node,0,num,max nsel,all ksel,all ksel,s,kp,,8 nslk,s *get,nkp8,node,0,num,max nsel,all ksel,all FINISH !************************************************** ********* !********************** SOLUTION ****************** ********* !************************************************** ********* /SOL

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47 Appendix A (Continued) ANTYPE,0 Specifies the a nalysis type and restart status and "0" means that it Performs a static analysis. Valid for all degree s of freedom NLGEOM,1 Includes largedeflection effects in a static or full transient analysis !************************************ DK,2, ,0, , ,UX,UY,UZ,ROTX,ROTZ, Boundary cond itions on keypoint 2 DK,3, ,0, , ,UX,UY,UZ,ROTX,ROTZ, Boundary cond itions on keypoint 2 DK,8, ,0, , ,UX,UY,UZ,ROTZ, LOCAL,11,CART,0,0,0,-44,0,0, CSYS,11 DK,8,ROTY,0 CSYS,0 !************************************ *DIM,my1,TABLE,10000 lsnum =0 *DO,step,1,141,10 theta=-1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO stop1 = lsnum *DO,step,141,156,1 theta=-1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO stop1 = lsnum DKDELE,2,ROTY

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48 Appendix A (Continued) *DO,step,-12450,47150,100 !Maximum Moment Value -2.10 65e+004 mmy1 = step*1 FK,2,MY,mmy1 lsnum=lsnum+1 *set,MY1(lsnum),step*1 LSWRITE,lsnum *ENDDO FKDELE,2,my *DO,step,47150,47250,10 !Maximum Moment Value -2.1065e+004 mmy1 = step*1 FK,2,MY,mmy1 lsnum=lsnum+1 *set,MY1(lsnum),step*1 LSWRITE,lsnum *ENDDO FKDELE,2,my *DO,step,16,180,0.5 theta=-1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO DKDELE,2,ROTY *Do,nn,1,lsnum LSSOLVE,nn /output,progress,txt,,append *VWRITE,nn Writes data to a file in a formatted sequence %16.8G

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49 Appendix A (Continued) /output *enddo /STATUS,SOLU FINISH !***************************** !******* GET RESULTS ********* !***************************** loadSteps=lsnum /POST1 *DIM,rotY2,TABLE,loadSteps *DIM,disX3,TABLE,loadSteps *DIM,disY3,TABLE,loadSteps *DIM,disZ3,TABLE,loadSteps *DIM,disX5,TABLE,loadSteps *DIM,disY5,TABLE,loadSteps *DIM,disZ5,TABLE,loadSteps *DIM,my2,TABLE,loadSteps /output,dofsoln,txt,,append Writes data to a file in a formatted sequence NLIST,ALL, , ,NODE,NODE,NODE /output *Do,nn,1,lsnum set,nn *GET,roty,Node,nkp2,ROT,Y *SET,rotY2(nn),roty *GET,mmy2,Node,nkp2,RF,MY *SET,my2(nn),mmy2 /output,dofsoln,txt,,append Writes data to a file in a formatted sequence PRNSOL,U,COMP /output *ENDDO /output,output_arc%arclength%_asp%aspect%,txt,, *MSG,INFO,'t','w','R','E','arclength' Wr ites an output message via the ANSYS message subroutine %-8C %-8C %-8C %-8C %-8C

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50 Appendix A (Continued) *VWRITE,h2,b2,R,E2,arclength Wr ites data to a file in a formatted sequence %16.8G %-16.8G %-16.8G %-16.8G %-16.8G *MSG,INFO,'roty2','my1','my2' %-8C %-8C %-8c *VWRITE,rotY2(1),my1(1),my2(1) %16.8G %-16.8G %-16.8G /output FINISH A.2 Batch File 2 !********** Create Keypoints 1 throug 7: K(Point #, X-Coord, Y-Coord, Z-Coord) ********* K,1,0,0,0, K,2,0,-225,0, K,3,0,-20,0, K,4,0,20,0, K,5,R*cos(PI/180*72),R*SIN(PI/180*72),0, K,6,R*cos(PI/180*33),R*sin(PI/180*33),0, K,7,R*cos(PI/180*29),R*sin(PI/180*29),0, K,8,R*cos(PI/180*44),-R*sin(PI/180*44),0, *DO,step,1,161,10 theta=-1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO stop1 = lsnum *DO,step,161,164,1 theta=-1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO

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51 Appendix A (Continued) stop1 = lsnum DKDELE,2,ROTY *DO,step,-13300,35000,100 mmy1 = step*1 FK,2,MY,mmy1 lsnum=lsnum+1 *set,MY1(lsnum),step*1 LSWRITE,lsnum *ENDDO FKDELE,2,my *DO,step,35000,35190,10 mmy1 = step*1 FK,2,MY,mmy1 lsnum=lsnum+1 *set,MY1(lsnum),step*1 LSWRITE,lsnum *ENDDO FKDELE,2,my *DO,step,31,180,1 theta=-1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO DKDELE,2,ROTY

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52 Appendix A (Continued) A.3 Batch File 3 !********** Create Keypoints 1 throug 7: K(Point #, X-Coord, Y-Coord, Z-Coord) ********* K,1,0,0,0, K,2,0,-225,0, K,3,0,-25,0, K,4,0,25,0, K,5,R*cos(PI/180*72),R*SIN(PI/180*72),0, K,6,R*cos(PI/180*33),R*sin(PI/180*33),0, K,7,R*cos(PI/180*29),R*sin(PI/180*29),0, K,8,R*cos(PI/180*44),-R*sin(PI/180*44),0, *DO,step,1,161,10 theta=-1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO stop1 = lsnum DKDELE,2,ROTY *DO,step,161,172,1 theta=-1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO stop1 = lsnum DKDELE,2,ROTY *DO,step,-14000,29000,100 !Maximum Moment Value -2.106 5e+004 mmy1 = step*1 FK,2,MY,mmy1 lsnum=lsnum+1 *set,MY1(lsnum),step*1

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53 Appendix A (Continued) LSWRITE,lsnum *ENDDO FKDELE,2,my *DO,step,35,180,1 theta=-1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO DKDELE,2,ROTY A.4 Batch File 4 !********** Create Keypoints 1 throug 7: K(Point #, X-Coord, Y-Coord, Z-Coord) ********* K,1,0,0,0, K,2,0,-225,0, K,3,0,-30,0, K,4,0,30,0, K,5,R*cos(PI/180*72),R*SIN(PI/180*72),0, K,6,R*cos(PI/180*33),R*sin(PI/180*33),0, K,7,R*cos(PI/180*29),R*sin(PI/180*29),0, K,8,R*cos(PI/180*44),-R*sin(PI/180*44),0, *DO,step,1,171,10 theta=-1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO stop1 = lsnum DKDELE,2,ROTY *DO,step,-18000,24900,100 !Maximum Moment Value -2.1065e+004 mmy1 = step*1

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54 Appendix A (Continued) FK,2,MY,mmy1 lsnum=lsnum+1 *set,MY1(lsnum),step*1 LSWRITE,lsnum *ENDDO FKDELE,2,my *DO,step,34,180,0.5 theta=-1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO A.5 Batch File 5 !********** Create Keypoints 1 throug 7: K(Point #, X-Coord, Y-Coord, Z-Coord) ********* K,1,0,0,0, K,2,0,-225,0, K,3,0,-40,0, K,4,0,40,0, K,5,R*cos(PI/180*72),R*SIN(PI/180*72),0, K,6,R*cos(PI/180*33),R*sin(PI/180*33),0, K,7,R*cos(PI/180*29),R*sin(PI/180*29),0, K,8,R*cos(PI/180*44),-R*sin(PI/180*44),0, *DO,step,1,181,10 theta=-1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO stop1 = lsnum DKDELE,2,ROTY *DO,step,181,196,1

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55 Appendix A (Continued) theta=-1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO stop1 = lsnum DKDELE,2,ROTY *DO,step,-14500,18100,100 !Maximum Moment Value -2.1065e+004 mmy1 = step*1 FK,2,MY,mmy1 lsnum=lsnum+1 *set,MY1(lsnum),step*1 LSWRITE,lsnum *ENDDO FKDELE,2,my *DO,step,18100,18140,10 !Maximum Moment Value -2.1065e+004 mmy1 = step*1 FK,2,MY,mmy1 lsnum=lsnum+1 *set,MY1(lsnum),step*1 LSWRITE,lsnum *ENDDO FKDELE,2,my *DO,step,53,27,-1 theta=-1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1

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56 Appendix A (Continued) LSWRITE,lsnum *ENDDO DKDELE,2,ROTY *DO,step,17500,19875,25 !Maximum Moment Value -2.10 65e+004 mmy1 = step*1 FK,2,MY,mmy1 lsnum=lsnum+1 *set,MY1(lsnum),step*1 LSWRITE,lsnum *ENDDO FKDELE,2,my *DO,step,40,180,1 theta=-1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO A.6 Batch File 6 !********** Create Keypoints 1 throug 7: K(Point #, X-Coord, Y-Coord, Z-Coord) ********* K,1,0,0,0, K,2,0,-225,0, K,3,0,-50,0, K,4,0,50,0, K,5,R*cos(PI/180*72),R*SIN(PI/180*72),0, K,6,R*cos(PI/180*33),R*sin(PI/180*33),0, K,7,R*cos(PI/180*29),R*sin(PI/180*29),0, K,8,R*cos(PI/180*44),-R*sin(PI/180*44),0, *DO,step,1,215,15 theta=-1*step DK,2,ROTY,theta*PI/180

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57 Appendix A (Continued) lsnum=lsnum+1 LSWRITE,lsnum *ENDDO stop1 = lsnum DKDELE,2,ROTY *DO,step,-15100,15900,100 !Maximum Moment Value -2.1065e+004 mmy1 = step*1 FK,2,MY,mmy1 lsnum=lsnum+1 *set,MY1(lsnum),step*1 LSWRITE,lsnum *ENDDO FKDELE,2,my *DO,step,-56,-25,1 theta=1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO DKDELE,2,ROTY *DO,step,12925,16975,25 !Maximum Moment Value -2.1065e+004 mmy1 = step*1 FK,2,MY,mmy1 lsnum=lsnum+1 *set,MY1(lsnum),step*1 LSWRITE,lsnum *ENDDO FKDELE,2,my *DO,step,-39,-180,-1

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58 Appendix A (Continued) theta=1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO A.7 Batch File 7 !********** Create Keypoints 1 throug 7: K(Point #, X-Coord, Y-Coord, Z-Coord) ********* K,1,0,0,0, K,2,0,-225,0, K,3,0,-60,0, K,4,0,60,0, K,5,R*cos(PI/180*72),R*SIN(PI/180*72),0, K,6,R*cos(PI/180*33),R*sin(PI/180*33),0, K,7,R*cos(PI/180*29),R*sin(PI/180*29),0, K,8,R*cos(PI/180*44),-R*sin(PI/180*44),0, *DO,step,1,231,10 theta=-1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO stop1 = lsnum !DKDELE,2,ROTY *DO,step,231,237,1 theta=-1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO stop1 = lsnum DKDELE,2,ROTY

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59 Appendix A (Continued) *DO,step,-14400,14600,100 !Maximum Moment Value -2.1065e+004 mmy1 = step*1 FK,2,MY,mmy1 lsnum=lsnum+1 *set,MY1(lsnum),step*1 LSWRITE,lsnum *ENDDO FKDELE,2,my *DO,step,14600,14790,10 !Maximum Moment Value -2.1065e+004 mmy1 = step*1 FK,2,MY,mmy1 lsnum=lsnum+1 *set,MY1(lsnum),step*1 LSWRITE,lsnum *ENDDO FKDELE,2,my *DO,step,-48,-24,1 theta=1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO DKDELE,2,ROTY *DO,step,10000,15150,25 !Maximum Moment Value -2.1065e+004 mmy1 = step*1 FK,2,MY,mmy1

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60 Appendix A (Continued) lsnum=lsnum+1 *set,MY1(lsnum),step*1 LSWRITE,lsnum *ENDDO FKDELE,2,my *DO,step,39,180,1 theta=-1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO

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61 Appendix B: MATLAB M-Files B.1 FEA and Twisting and Bending Plots This MATLAB file selects the correct data from the pre vious FEA results and plots it against the twisting and bending model results also produced by this M-file. clc input_range=[pi/2:pi/288:3*pi/2]; %--------------------------------------------------------------------%spherical trig wrt theta2 as input r1=46*pi/180; %spherical trig angle r1=46degrees r2=pi/2; %spherical trig angle r2=90degrees r3=pi/2; %spherical trig angle r3=90degrees r4=75*pi/180; %spherical trig angle r4=75degrees omega = input_range; %Input theta2 flag1 = sign(sin(omega)); delta_wrt2 = acos(cos(r1)*cos(r2)+sin(r1)*sin(r2)*c os(omega)); alpha_wrt2 = flag1.*(acos((cos(r1)-cos(r2)*cos(delta_wrt2))./(sin(r2)*sin(delta_wrt2)) )); beta_wrt2 = acos((cos(r4)-cos(r3)*cos(delta_wrt2))./(sin(r3)*sin(delta_wrt2)) ); gamma_wrt2 = acos((cos(delta_wrt2)-cos(r3)*cos(r4))./(sin(r3)*sin(r4))); epsilon_wrt2 = acos((cos(r3)-cos(r4)*cos(delta_wrt2))./(sin(r3)*sin(delta_wrt2)) ); zeta_wrt2 = flag1.*(acos((cos(r2)-cos(r1)*cos(delta_wrt2))./(sin(r1)*sin(delta_wrt2)) )); theta4_wrt2 = pi-epsilon_wrt2-zeta_wrt2; %--------------------------------------------------------------------E = 169*10^3; %youngs modulus 169 GPa a = 3.5; %width of compliant joint b = 2; %height of compliant joint omega_match_ansys = -(omega-pi/2); %moment of bending joint3 L3 = 22.17; %length of joint 3 in micrometers Ix = a*b^3/12; %second moment of area, compliant mechanisms pg.399 K3 = -E*Ix/L3; %bending coefficient

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62 Appendix B (Continued) delta_psi3 = (gamma_wrt2-gamma_wrt2(1)); %need to make sure this gets smaller d_Phi3_d_omega=(sin(r1).*sin(r2).*sin(omega))./(sin (r3).*sin(r4).*sin(g amma_wrt2)); %U_bending_J3 = 0.5*K3*(delta_psi3).^2; MJ3 = K3.*delta_psi3.*d_Phi3_d_omega; %----------------------------------------------------------------------%1square Analytical L=20; %coefficient of twist K1 nu = 0.35; %poisson's ratio beta = 0.2125; %pg 255 elasticity J = beta*a*b^3; %J is polar moment of inertia, JG is the torsional rigidity, pg. 191 compliant mechanisms G = E/(2*(1+nu)); %shear modulus K1 = -J*G./L; %coefficient of twist %moment of bending joint2. Ix = a*b^3/12; %second moment of area, compliant mechanisms pg. 399 K2 = -E*Ix./L; %bending coefficient delta_phi2 = (alpha_wrt2(1)-alpha_wrt2)+(beta_wrt2( 1)-beta_wrt2); d_alpha_d_omega=((sin(r1).*sin(omega))./(sin(alpha_ wrt2).*sin(delta_wrt 2).^3)).*(cos(r1).*cos(delta_wrt2)-cos(r2).*cos(del ta_wrt2).^2cos(r2).*sin(delta_wrt2).^2); d_beta_d_omega=((sin(r1).*sin(r2).*sin(omega))./(si n(beta_wrt2).*sin(r3 ).*sin(delta_wrt2).^3)).*(cos(r4).*cos(delta_wrt2)-cos(r3).*sin(delta_wrt2).^2-cos(r3).*cos(delta_wrt2 ).^2); d_phi2_d_omega=d_alpha_d_omega+d_beta_d_omega; %U_bending_J3 = 0.5*K2*(delta_psi2).^2; MJ2 = K2*delta_phi2.*d_phi2_d_omega; theta2_1 = omega_match_ansys-1/K1.*(MJ2+MJ3); M2_1=(MJ2+MJ3); M2_1(145)=(M2_1(144)+M2_1(146))/2; theta2_1(145)=(theta2_1(144)+theta2_1(146))/2; % 1 Square Ansys filename = [ 'ansys_output_1square.txt' ]; string1 = [ 'I:\Thesis\Chester\Thesis Data\Ansys\Combined Data\ ]; fid1 = fopen([string1,filename]); % opens the file ABT = fread(fid1); % reads the file into variable ABT fclose(fid1); % closes the data file GBT = native2unicode(ABT)'; % changes data from machine code to text s_iB = findstr( 'my2' ,GBT); % finds end of header A=str2num(GBT(s_iB+4:end)); % turns the data into a numerical matrix

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63 Appendix B (Continued) roty2_1 = [0; A(:,1)]; % pulls roty2 from file and adds a zero to beginning my2 = [0; A(:,2)]; % pulls my2 from file and adds a zero to beginning my3 = [0; A(:,3)]; % pulls my3 from file and adds a zero to beginning flag2 = abs(sign(my2)); % assigns a zero to values of zero and 1 to values other than zero my_1 = (flag2.*my2+(1-flag2).*my3); %combines my2 and my3 by using previous 1's and zeros to form one column,' my %----------------------------------------------------------------------%2square Analytical L=30; %coefficient of twist K1 nu = 0.35; %poisson's ratio beta = 0.2125; %pg 255 elasticity J = beta*a*b^3; %J is polar moment of inertia, JG is the torsional rigidity, pg. 191 compliant mechanisms G = E/(2*(1+nu)); %shear modulus K1 = -J*G./L; %coefficient of twist %moment of bending joint2. Ix = a*b^3/12; %second moment of area, compliant mechanisms pg. 399 K2 = -E*Ix./L; %bending coefficient delta_phi2 = (alpha_wrt2(1)-alpha_wrt2)+(beta_wrt2( 1)-beta_wrt2); d_alpha_d_omega=((sin(r1).*sin(omega))./(sin(alpha_ wrt2).*sin(delta_wrt 2).^3)).*(cos(r1).*cos(delta_wrt2)-cos(r2).*cos(del ta_wrt2).^2cos(r2).*sin(delta_wrt2).^2); d_beta_d_omega=((sin(r1).*sin(r2).*sin(omega))./(si n(beta_wrt2).*sin(r3 ).*sin(delta_wrt2).^3)).*(cos(r4).*cos(delta_wrt2)-cos(r3).*sin(delta_wrt2).^2-cos(r3).*cos(delta_wrt2 ).^2); d_phi2_d_omega=d_alpha_d_omega+d_beta_d_omega; %U_bending_J3 = 0.5*K2*(delta_psi2).^2; MJ2 = K2*delta_phi2.*d_phi2_d_omega; theta2_2 = omega_match_ansys-1/K1.*(MJ2+MJ3); M2_2=(MJ2+MJ3); M2_2(145)=(M2_2(144)+M2_2(146))/2; theta2_2(145)=(theta2_2(144)+theta2_2(146))/2; % 2 Square Ansys filename = [ 'ansys_output_2square.txt' ]; string1 = [ 'I:\Thesis\Chester\Thesis Data\Ansys\Combined Data\ ]; fid1 = fopen([string1,filename]); % opens the file ABT = fread(fid1); % reads the file into variable ABT fclose(fid1); % closes the data file

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64 Appendix B (Continued) GBT = native2unicode(ABT)'; % changes data from machine code to text s_iB = findstr( 'my2' ,GBT); % finds end of header A=str2num(GBT(s_iB+4:end)); % turns the data into a numerical matrix roty2_2 = [0; A(:,1)]; % pulls roty2 from file and adds a zero to beginning my2 = [0; A(:,2)]; % pulls my2 from file and adds a zero to beginning my3 = [0; A(:,3)]; % pulls my3 from file and adds a zero to beginning flag2 = abs(sign(my2)); % assigns a zero to values of zero and 1 to values other than zero my_2 = (flag2.*my2+(1-flag2).*my3); %combines my2 and my3 by using previous 1's and zeros to form one column,' my %----------------------------------------------------------------------%3square Analytical L=40; %coefficient of twist K1 nu = 0.35; %poisson's ratio beta = 0.2125; %pg 255 elasticity J = beta*a*b^3; %J is polar moment of inertia, JG is the torsional rigidity, pg. 191 compliant mechanisms G = E/(2*(1+nu)); %shear modulus K1 = -J*G./L; %coefficient of twist %moment of bending joint2. Ix = a*b^3/12; %second moment of area, compliant mechanisms pg. 399 K2 = -E*Ix./L; %bending coefficient delta_phi2 = (alpha_wrt2(1)-alpha_wrt2)+(beta_wrt2( 1)-beta_wrt2);; d_alpha_d_omega=((sin(r1).*sin(omega))./(sin(alpha_ wrt2).*sin(delta_wrt 2).^3)).*(cos(r1).*cos(delta_wrt2)-cos(r2).*cos(del ta_wrt2).^2cos(r2).*sin(delta_wrt2).^2); d_beta_d_omega=((sin(r1).*sin(r2).*sin(omega))./(si n(beta_wrt2).*sin(r3 ).*sin(delta_wrt2).^3)).*(cos(r4).*cos(delta_wrt2)-cos(r3).*sin(delta_wrt2).^2-cos(r3).*cos(delta_wrt2 ).^2); d_phi2_d_omega=d_alpha_d_omega+d_beta_d_omega; %U_bending_J3 = 0.5*K2*(delta_psi2).^2; MJ2 = K2*delta_phi2.*d_phi2_d_omega; theta2_3 = omega_match_ansys-1/K1.*(MJ2+MJ3); M2_3=(MJ2+MJ3); M2_3(145)=(M2_3(144)+M2_3(146))/2; theta2_3(145)=(theta2_3(144)+theta2_3(146))/2; % 3 Square Ansys filename = [ 'ansys_output_3square.txt' ]; string1 = [ 'I:\Thesis\Chester\Thesis Data\Ansys\Combined Data\ ];

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65 Appendix B (Continued) fid1 = fopen([string1,filename]); % opens the file ABT = fread(fid1); % reads the file into variable ABT fclose(fid1); % closes the data file GBT = native2unicode(ABT)'; % changes data from machine code to text s_iB = findstr( 'my2' ,GBT); % finds end of header A=str2num(GBT(s_iB+4:end)); % turns the data into a numerical matrix roty2_3 = [0; A(:,1)]; % pulls roty2 from file and adds a zero to beginning my2 = [0; A(:,2)]; % pulls my2 from file and adds a zero to beginning my3 = [0; A(:,3)]; % pulls my3 from file and adds a zero to beginning flag2 = abs(sign(my2)); % assigns a zero to values of zero and 1 to values other than zero my_3 = (flag2.*my2+(1-flag2).*my3); %combines my2 and my3 by using previous 1's and zeros to form one column,' my my_3(156)=0; % deletes incorrect data in 3square program to remo ve spike %----------------------------------------------------------------------%4square Analytical L=50; %coefficient of twist K1 nu = 0.35; %poisson's ratio beta = 0.2125; %pg 255 elasticity J = beta*a*b^3; %J is polar moment of inertia, JG is the torsional rigidity, pg. 191 compliant mechanisms G = E/(2*(1+nu)); %shear modulus K1 = -J*G./L; %coefficient of twist %moment of bending joint2. Ix = a*b^3/12; %second moment of area, compliant mechanisms pg. 399 K2 = -E*Ix./L; %bending coefficient delta_phi2 = (alpha_wrt2(1)-alpha_wrt2)+(beta_wrt2( 1)-beta_wrt2); d_alpha_d_omega=((sin(r1).*sin(omega))./(sin(alpha_ wrt2).*sin(delta_wrt 2).^3)).*(cos(r1).*cos(delta_wrt2)-cos(r2).*cos(del ta_wrt2).^2cos(r2).*sin(delta_wrt2).^2); d_beta_d_omega=((sin(r1).*sin(r2).*sin(omega))./(si n(beta_wrt2).*sin(r3 ).*sin(delta_wrt2).^3)).*(cos(r4).*cos(delta_wrt2)-cos(r3).*sin(delta_wrt2).^2-cos(r3).*cos(delta_wrt2 ).^2); d_phi2_d_omega=d_alpha_d_omega+d_beta_d_omega; %U_bending_J3 = 0.5*K2*(delta_psi2).^2; MJ2 = K2*delta_phi2.*d_phi2_d_omega; theta2_4 = omega_match_ansys-1/K1.*(MJ2+MJ3); M2_4=(MJ2+MJ3);

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66 Appendix B (Continued) M2_4(145)=(M2_4(144)+M2_4(146))/2; theta2_4(145)=(theta2_4(144)+theta2_4(146))/2; % 4 Square Ansys filename = [ 'ansys_output_4square.txt' ]; string1 = [ 'I:\Thesis\Chester\Thesis Data\Ansys\Combined Data\ ]; fid1 = fopen([string1,filename]); % opens the file ABT = fread(fid1); % reads the file into variable ABT fclose(fid1); % closes the data file GBT = native2unicode(ABT)'; % changes data from machine code to text s_iB = findstr( 'my2' ,GBT); % finds end of header A=str2num(GBT(s_iB+4:end)); % turns the data into a numerical matrix roty2_4 = [0; A(:,1)]; % pulls roty2 from file and adds a zero to beginning my2 = [0; A(:,2)]; % pulls my2 from file and adds a zero to beginning my3 = [0; A(:,3)]; % pulls my3 from file and adds a zero to beginning flag2 = abs(sign(my2)); % assigns a zero to values of zero and 1 to values other than zero my_4 = (flag2.*my2+(1-flag2).*my3); %combines my2 and my3 by using previous 1's and zeros to form one column,' my my_4(171)=0; % deletes incorrect data in 4square program to remo ve spike %----------------------------------------------------------------------%5square Analytical L=60; %coefficient of twist K1 nu = 0.35; %poisson's ratio beta = 0.2125; %pg 255 elasticity J = beta*a*b^3; %J is polar moment of inertia, JG is the torsional rigidity, pg. 191 compliant mechanisms G = E/(2*(1+nu)); %shear modulus K1 = -J*G./L; %coefficient of twist %moment of bending joint2. Ix = a*b^3/12; %second moment of area, compliant mechanisms pg. 399 K2 = -E*Ix./L; %bending coefficient delta_phi2 = (alpha_wrt2(1)-alpha_wrt2)+(beta_wrt2( 1)-beta_wrt2); d_alpha_d_omega=((sin(r1).*sin(omega))./(sin(alpha_ wrt2).*sin(delta_wrt 2).^3)).*(cos(r1).*cos(delta_wrt2)-cos(r2).*cos(del ta_wrt2).^2cos(r2).*sin(delta_wrt2).^2); d_beta_d_omega=((sin(r1).*sin(r2).*sin(omega))./(si n(beta_wrt2).*sin(r3 ).*sin(delta_wrt2).^3)).*(cos(r4).*cos(delta_wrt2)-cos(r3).*sin(delta_wrt2).^2-cos(r3).*cos(delta_wrt2 ).^2); d_phi2_d_omega=d_alpha_d_omega+d_beta_d_omega; %U_bending_J3 = 0.5*K2*(delta_psi2).^2;

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67 Appendix B (Continued) MJ2 = K2*delta_phi2.*d_phi2_d_omega; theta2_5 = omega_match_ansys-1/K1.*(MJ2+MJ3); M2_5=(MJ2+MJ3); M2_5(145)=(M2_5(144)+M2_5(146))/2; theta2_5(145)=(theta2_5(144)+theta2_5(146))/2; % 5 Square Ansys filename = [ 'ansys_output_5square.txt' ]; string1 = [ 'I:\Thesis\Chester\Thesis Data\Ansys\Combined Data\ ]; fid1 = fopen([string1,filename]); % opens the file ABT = fread(fid1); % reads the file into variable ABT fclose(fid1); % closes the data file GBT = native2unicode(ABT)'; % changes data from machine code to text s_iB = findstr( 'my2' ,GBT); % finds end of header A=str2num(GBT(s_iB+4:end)); % turns the data into a numerical matrix roty2_5 = [0; A(:,1)]; % pulls roty2 from file and adds a zero to beginning my2 = [0; A(:,2)]; % pulls my2 from file and adds a zero to beginning my3 = [0; A(:,3)]; % pulls my3 from file and adds a zero to beginning flag2 = abs(sign(my2)); % assigns a zero to values of zero and 1 to values other than zero my_5 = (flag2.*my2+(1-flag2).*my3); %combines my2 and my3 by using previous 1's and zeros to form one column,' my my_5(200)=0; % deletes incorrect data in 5square program to remo ve spike %----------------------------------------------------------------------%6square Analytical L=80; %coefficient of twist K1 nu = 0.35; %poisson's ratio beta = 0.2125; %pg 255 elasticity J = beta*a*b^3; %J is polar moment of inertia, JG is the torsional rigidity, pg. 191 compliant mechanisms G = E/(2*(1+nu)); %shear modulus K1 = -J*G./L; %coefficient of twist %moment of bending joint2. Ix = a*b^3/12; %second moment of area, compliant mechanisms pg. 399 K2 = -E*Ix./L; %bending coefficient delta_phi2 = (alpha_wrt2(1)-alpha_wrt2)+(beta_wrt2( 1)-beta_wrt2);

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68 Appendix B (Continued) d_alpha_d_omega=((sin(r1).*sin(omega))./(sin(alpha_ wrt2).*sin(delta_wrt 2).^3)).*(cos(r1).*cos(delta_wrt2)-cos(r2).*cos(del ta_wrt2).^2cos(r2).*sin(delta_wrt2).^2); d_beta_d_omega=((sin(r1).*sin(r2).*sin(omega))./(si n(beta_wrt2).*sin(r3 ).*sin(delta_wrt2).^3)).*(cos(r4).*cos(delta_wrt2)-cos(r3).*sin(delta_wrt2).^2-cos(r3).*cos(delta_wrt2 ).^2); d_phi2_d_omega=d_alpha_d_omega+d_beta_d_omega; %U_bending_J3 = 0.5*K2*(delta_psi2).^2; MJ2 = K2*delta_phi2.*d_phi2_d_omega; theta2_6 = omega_match_ansys-1/K1.*(MJ2+MJ3); M2_6=(MJ2+MJ3); M2_6(145)=(M2_6(144)+M2_6(146))/2; theta2_6(145)=(theta2_6(144)+theta2_6(146))/2; % 6 Square Ansys filename = [ 'ansys_output_6square.txt' ]; string1 = [ 'I:\Thesis\Chester\Thesis Data\Ansys\Combined Data\ ]; fid1 = fopen([string1,filename]); % opens the file ABT = fread(fid1); % reads the file into variable ABT fclose(fid1); % closes the data file GBT = native2unicode(ABT)'; % changes data from machine code to text s_iB = findstr( 'my2' ,GBT); % finds end of header A=str2num(GBT(s_iB+4:end)); % turns the data into a numerical matrix roty2_6 = [0; A(:,1)]; % pulls roty2 from file and adds a zero to beginning my2 = [0; A(:,2)]; % pulls my2 from file and adds a zero to beginning my3 = [0; A(:,3)]; % pulls my3 from file and adds a zero to beginning flag2 = abs(sign(my2)); % assigns a zero to values of zero and 1 to values other than zero my_6 = (flag2.*my2+(1-flag2).*my3); %combines my2 and my3 by using previous 1's and zeros to form one column,' my my_6(182)=0; % deletes incorrect data in 6square program to remo ve spike %----------------------------------------------------------------------%7square Analytical L=100; %coefficient of twist K1 nu = 0.35; %poisson's ratio beta = 0.2125; %pg 255 elasticity J = beta*a*b^3; %J is polar moment of inertia, JG is the torsional rigidity, pg. 191 compliant mechanisms G = E/(2*(1+nu)); %shear modulus K1 = -J*G./L; %coefficient of twist

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69 Appendix B (Continued) %moment of bending joint2. Ix = a*b^3/12; %second moment of area, compliant mechanisms pg. 399 K2 = -E*Ix./L; %bending coefficient delta_phi2 = (alpha_wrt2(1)-alpha_wrt2)+(beta_wrt2( 1)-beta_wrt2); d_alpha_d_omega=((sin(r1).*sin(omega))./(sin(alpha_ wrt2).*sin(delta_wrt 2).^3)).*(cos(r1).*cos(delta_wrt2)-cos(r2).*cos(del ta_wrt2).^2cos(r2).*sin(delta_wrt2).^2); d_beta_d_omega=((sin(r1).*sin(r2).*sin(omega))./(si n(beta_wrt2).*sin(r3 ).*sin(delta_wrt2).^3)).*(cos(r4).*cos(delta_wrt2)-cos(r3).*sin(delta_wrt2).^2-cos(r3).*cos(delta_wrt2 ).^2); d_phi2_d_omega=d_alpha_d_omega+d_beta_d_omega; %U_bending_J3 = 0.5*K2*(delta_psi2).^2; MJ2 = K2*delta_phi2.*d_phi2_d_omega; theta2_7 = omega_match_ansys-1/K1.*(MJ2+MJ3); M2_7=(MJ2+MJ3); M2_7(145)=(M2_7(144)+M2_7(146))/2; theta2_7(145)=(theta2_7(144)+theta2_7(146))/2; % 7 Square Ansys filename = [ 'ansys_output_7square.txt' ]; string1 = [ 'I:\Thesis\Chester\Thesis Data\Ansys\Combined Data\ ]; fid1 = fopen([string1,filename]); % opens the file ABT = fread(fid1); % reads the file into variable ABT fclose(fid1); % closes the data file GBT = native2unicode(ABT)'; % changes data from machine code to text s_iB = findstr( 'my2' ,GBT); % finds end of header A=str2num(GBT(s_iB+4:end)); % turns the data into a numerical matrix roty2_7 = [0; A(:,1)]; % pulls roty2 from file and adds a zero to beginning my2 = [0; A(:,2)]; % pulls my2 from file and adds a zero to beginning my3 = [0; A(:,3)]; % pulls my3 from file and adds a zero to beginning flag2 = abs(sign(my2)); % assigns a zero to values of zero and 1 to values other than zero my_7 = (flag2.*my2+(1-flag2).*my3); %combines my2 and my3 by using previous 1's and zeros to form one column,' my my_7(168)=0; % deletes incorrect data in 7square program to remo ve spike %----------------------------------------------------------------------%8square Analytical L=120; %coefficient of twist K1

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70 Appendix B (Continued) nu = 0.35; %poisson's ratio beta = 0.2125; %pg 255 elasticity J = beta*a*b^3; %J is polar moment of inertia, JG is the torsional rigidity, pg. 191 compliant mechanisms G = E/(2*(1+nu)); %shear modulus K1 = -J*G./L; %coefficient of twist %moment of bending joint2. Ix = a*b^3/12; %second moment of area, compliant mechanisms pg. 399 K2 = -E*Ix./L; %bending coefficient delta_phi2 = (alpha_wrt2(1)-alpha_wrt2)+(beta_wrt2( 1)-beta_wrt2); d_alpha_d_omega=((sin(r1).*sin(omega))./(sin(alpha_ wrt2).*sin(delta_wrt 2).^3)).*(cos(r1).*cos(delta_wrt2)-cos(r2).*cos(del ta_wrt2).^2cos(r2).*sin(delta_wrt2).^2); d_beta_d_omega=((sin(r1).*sin(r2).*sin(omega))./(si n(beta_wrt2).*sin(r3 ).*sin(delta_wrt2).^3)).*(cos(r4).*cos(delta_wrt2)-cos(r3).*sin(delta_wrt2).^2-cos(r3).*cos(delta_wrt2 ).^2); d_phi2_d_omega=d_alpha_d_omega+d_beta_d_omega; %U_bending_J3 = 0.5*K2*(delta_psi2).^2; MJ2 = K2*delta_phi2.*d_phi2_d_omega; theta2_8 = omega_match_ansys-1/K1.*(MJ2+MJ3); M2_8=(MJ2+MJ3); M2_8(145)=(M2_8(144)+M2_8(146))/2; theta2_8(145)=(theta2_8(144)+theta2_8(146))/2; % 8 Square Ansys filename = [ 'ansys_output_8square.txt' ]; string1 = [ 'I:\Thesis\Chester\Thesis Data\Ansys\Combined Data\ ]; fid1 = fopen([string1,filename]); % opens the file ABT = fread(fid1); % reads the file into variable ABT fclose(fid1); % closes the data file GBT = native2unicode(ABT)'; % changes data from machine code to text s_iB = findstr( 'my2' ,GBT); % finds end of header A=str2num(GBT(s_iB+4:end)); % turns the data into a numerical matrix roty2_8 = [0; A(:,1)]; % pulls roty2 from file and adds a zero to beginning my2 = [0; A(:,2)]; % pulls my2 from file and adds a zero to beginning my3 = [0; A(:,3)]; % pulls my3 from file and adds a zero to beginning flag2 = abs(sign(my2)); % assigns a zero to values of zero and 1 to values other than zero my_8 = (flag2.*my2+(1-flag2).*my3); %combines my2 and my3 by using previous 1's and zeros to form one column,' my my_8(177)=0; % deletes incorrect data in 8square program to remo ve spike %----------------------------------------------------------------------

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71 Appendix B (Continued) %individual plots of ansys vs analytical figure(1) clf hold on set(gca, 'fontsize' ,15) xlabel( '\theta_2 (Degrees)' 'fontsize' ,17) ylabel( 'Moment (\muN\mum)' 'fontsize' ,17) h=[plot(roty2_1*180/pi,my_1, 'k' ) plot(theta2_1*180/pi,M2_1, 'b' )]; p=legend(h, 'ANSYS' 'Analytical' ); set(p, 'box' 'off' ) set(p, 'location' 'northwest' ) set(p, 'fontsize' ,14) print(gcf, '-dtiff' '-r300' 'Graph_MomVsRot_combined_1' ) figure(2) clf hold on set(gca, 'fontsize' ,15) xlabel( '\theta_2 (Degrees)' 'fontsize' ,17) ylabel( 'Moment (\muN\mum)' 'fontsize' ,17) h=[plot(roty2_2*180/pi,my_2, 'k' ) plot(theta2_2*180/pi,M2_2, 'b' )]; p=legend(h, 'ANSYS' 'Analytical' ); set(p, 'box' 'off' ) set(p, 'location' 'northwest' ) set(p, 'fontsize' ,14) print(gcf, '-dtiff' '-r300' 'Graph_MomVsRot_combined_2' ) figure(3) clf hold on set(gca, 'fontsize' ,15) xlabel( '\theta_2 (Degrees)' 'fontsize' ,17) ylabel( 'Moment (\muN\mum)' 'fontsize' ,17) h=[plot(roty2_3*180/pi,my_3, 'k' ) plot(theta2_3*180/pi,M2_3, 'b' )]; p=legend(h, 'ANSYS' 'Analytical' ); set(p, 'box' 'off' ) set(p, 'location' 'northwest' ) set(p, 'fontsize' ,14) print(gcf, '-dtiff' '-r300' 'Graph_MomVsRot_combined_3' ) figure(4) clf hold on set(gca, 'fontsize' ,15) xlabel( '\theta_2 (Degrees)' 'fontsize' ,17) ylabel( 'Moment (\muN\mum)' 'fontsize' ,17) h=[plot(roty2_4*180/pi,my_4, 'k' ) plot(theta2_4*180/pi,M2_4, 'b' )]; p=legend(h, 'ANSYS' 'Analytical' ); set(p, 'box' 'off' ) set(p, 'location' 'northwest' ) set(p, 'fontsize' ,14)

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72 Appendix B (Continued) print(gcf, '-dtiff' '-r300' 'Graph_MomVsRot_combined_4' ) figure(5) clf hold on set(gca, 'fontsize' ,15) xlabel( '\theta_2 (Degrees)' 'fontsize' ,17) ylabel( 'Moment (\muN\mum)' 'fontsize' ,17) h=[plot(roty2_5*180/pi,my_5, 'k' ) plot(theta2_5*180/pi,M2_5, 'b' )]; p=legend(h, 'ANSYS' 'Analytical' ); set(p, 'box' 'off' ) set(p, 'location' 'northwest' ) set(p, 'fontsize' ,14) print(gcf, '-dtiff' '-r300' 'Graph_MomVsRot_combined_5' ) figure(6) clf hold on set(gca, 'fontsize' ,15) xlabel( '\theta_2 (Degrees)' 'fontsize' ,17) ylabel( 'Moment (\muN\mum)' 'fontsize' ,17) h=[plot(roty2_6*180/pi,my_6, 'k' ) plot(theta2_6*180/pi,M2_6, 'b' )]; p=legend(h, 'ANSYS' 'Analytical' ); set(p, 'box' 'off' ) set(p, 'location' 'northwest' ) set(p, 'fontsize' ,14) print(gcf, '-dtiff' '-r300' 'Graph_MomVsRot_combined_6' ) figure(7) clf hold on set(gca, 'fontsize' ,15) xlabel( '\theta_2 (Degrees)' 'fontsize' ,17) ylabel( 'Moment (\muN\mum)' 'fontsize' ,17) h=[plot(roty2_7*180/pi,my_7, 'k' ) plot(theta2_7*180/pi,M2_7, 'b' )]; p=legend(h, 'ANSYS' 'Analytical' ); set(p, 'box' 'off' ) set(p, 'location' 'northwest' ) set(p, 'fontsize' ,14) print(gcf, '-dtiff' '-r300' 'Graph_MomVsRot_combined_7' ) figure(8) clf hold on set(gca, 'fontsize' ,15) xlabel( '\theta_2 (Degrees)' 'fontsize' ,17) ylabel( 'Moment (\muN\mum)' 'fontsize' ,17) h=[plot(roty2_8*180/pi,my_8, 'k' ) plot(theta2_8*180/pi,M2_8, 'b' )]; p=legend(h, 'ANSYS' 'Analytical' ); set(p, 'box' 'off' ) set(p, 'location' 'northwest' ) set(p, 'fontsize' ,14) print(gcf, '-dtiff' '-r300' 'Graph_MomVsRot_combined_8' )

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73 Appendix B (Continued) % %----------------------------------------------------------------------%all combined plots %combined plot for models 1-8 figure(9) clf hold on set(gca, 'fontsize' ,15) xlabel( '\theta_2 (Degrees)' 'fontsize' ,17) ylabel( 'Moment (\muN\mum)' 'fontsize' ,17) thinline=plot(roty2_2*180/pi,my_2, 'k' ); thickline=plot(theta2_2*180/pi,M2_2, 'k' 'linewidth' ,3); plot(theta2_1*180/pi,M2_1, 'k' 'linewidth' ,3) plot(theta2_2*180/pi,M2_2, 'r' 'linewidth' ,3) plot(theta2_3*180/pi,M2_3, 'm' 'linewidth' ,3) plot(theta2_4*180/pi,M2_4, 'b' 'linewidth' ,3) plot(theta2_5*180/pi,M2_5, 'g' 'linewidth' ,3) plot(theta2_6*180/pi,M2_6, 'c' 'linewidth' ,3) plot(theta2_7*180/pi,M2_7, 'k' 'linewidth' ,3) plot(theta2_8*180/pi,M2_8, 'r' 'linewidth' ,3) h=[plot(theta2_1(1)*180/pi,M2_1(1), '*' ) thinline thickline plot(roty2_1*180/pi,my_1, 'k' ) plot(roty2_2*180/pi,my_2, 'r' ) plot(roty2_3*180/pi,my_3, 'm' ) plot(roty2_4*180/pi,my_4, 'b' ) plot(roty2_5*180/pi,my_5, 'g' ) plot(roty2_6*180/pi,my_6, 'c' ) plot(roty2_7*180/pi,my_7, 'k' ) plot(roty2_8*180/pi,my_8, 'r' )]; p=legend(h, 'Start' 'Ansys' 'Analytical' 'Model 1' 'Model 2' 'Model 3' 'Model 4' 'Model 5' 'Model 6' 'Model 7' 'Model 8' ,2); set(p, 'box' 'off' ) set(p, 'location' 'northeast' ) set(p, 'fontsize' ,10) print(gcf, '-dtiff' '-r300' 'Graph_MomVsRot_Allcombined_2-8' ) %combined plot for models 2-8 figure(10) clf hold on set(gca, 'fontsize' ,15) xlabel( '\theta_2 (Degrees)' 'fontsize' ,17) ylabel( 'Moment (\muN\mum)' 'fontsize' ,17) thinline=plot(roty2_2*180/pi,my_2, 'k' ); thickline=plot(theta2_2*180/pi,M2_2, 'k' 'LineWidth' ,3); plot(theta2_2*180/pi,M2_2, 'k' 'LineWidth' ,3) plot(theta2_3*180/pi,M2_3, 'r' 'LineWidth' ,3) plot(theta2_4*180/pi,M2_4, 'm' 'LineWidth' ,3) plot(theta2_5*180/pi,M2_5, 'b' 'LineWidth' ,3) plot(theta2_6*180/pi,M2_6, 'g' 'LineWidth' ,3)

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74 Appendix B (Continued) plot(theta2_7*180/pi,M2_7, 'c' 'LineWidth' ,3) plot(theta2_8*180/pi,M2_8, 'k' 'LineWidth' ,3) h=[plot(theta2_1(1)*180/pi,M2_1(1), '*' ) thinline thickline plot(roty2_2*180/pi,my_2, 'k' ) plot(roty2_3*180/pi,my_3, 'r' ) plot(roty2_4*180/pi,my_4, 'm' ) plot(roty2_5*180/pi,my_5, 'b' ) plot(roty2_6*180/pi,my_6, 'g' ) plot(roty2_7*180/pi,my_7, 'c' ) plot(roty2_8*180/pi,my_8, 'k' )]; p=legend(h, 'Start' 'Ansys' 'Analytical' 'Model 2' 'Model 3' 'Model 4' 'Model 5' 'Model 6' 'Model 7' 'Model 8' ,2); set(p, 'box' 'off' ) set(p, 'location' 'northwest' ) set(p, 'fontsize' ,10) print(gcf, '-dtiff' '-r300' 'Graph_MomVsRot_Allcombined_2-8' ) B.2 FEA and Pure Bending Plots The following plots the pure bending model against FE A results. clc input_range=[pi/2:pi/288:3*pi/2]; %----------------------------------------------------------------------%spherical trig wrt theta2 as input r1=46*pi/180; %spherical trig angle r1=46degrees r2=pi/2; %spherical trig angle r2=90degrees r3=pi/2; %spherical trig angle r3=90degrees r4=75*pi/180; %spherical trig angle r4=75degrees omega = input_range; %Input theta2 flag1 = sign(sin(omega)); delta_wrt2 = acos(cos(r1)*cos(r2)+sin(r1)*sin(r2)*c os(omega)); alpha_wrt2 = flag1.*(acos((cos(r1)-cos(r2)*cos(delta_wrt2))./(sin(r2)*sin(delta_wrt2)) )); beta_wrt2 = acos((cos(r4)-cos(r3)*cos(delta_wrt2))./(sin(r3)*sin(delta_wrt2)) ); gamma_wrt2 = acos((cos(delta_wrt2)-cos(r3)*cos(r4))./(sin(r3)*sin(r4))); epsilon_wrt2 = acos((cos(r3)-cos(r4)*cos(delta_wrt2))./(sin(r3)*sin(delta_wrt2)) );

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75 Appendix B (Continued) zeta_wrt2 = flag1.*(acos((cos(r2)-cos(r1)*cos(delta_wrt2))./(sin(r1)*sin(delta_wrt2)) )); theta4_wrt2 = pi-epsilon_wrt2-zeta_wrt2; %----------------------------------------------------------------------E = 169*10^3; %youngs modulus 169 GPa a = 3.5; %width of compliant joint b = 2; %height of compliant joint omega_match_ansys = -(omega-pi/2); %moment of bending joint3 L3 = 22.17; %length of joint 3 in micrometers Ix = a*b^3/12; %second moment of area, compliant mechanisms pg.399 K3 = -E*Ix/L3; %bending coefficient delta_psi3 = (gamma_wrt2-gamma_wrt2(1)); %need to make sure this gets smaller d_Phi3_d_omega=(sin(r1).*sin(r2).*sin(omega))./(sin (r3).*sin(r4).*sin(g amma_wrt2)); %U_bending_J3 = 0.5*K3*(delta_psi3).^2; MJ3 = K3.*delta_psi3.*d_Phi3_d_omega; %----------------------------------------------------------------------%1square Analytical L=20; %moment of bending joint2. Ix = a*b^3/12; %second moment of area, compliant mechanisms pg. 399 K2 = -E*Ix./L; %bending coefficient delta_phi2 = (alpha_wrt2(1)-alpha_wrt2)+(beta_wrt2( 1)-beta_wrt2); d_alpha_d_omega=((sin(r1).*sin(omega))./(sin(alpha_ wrt2).*sin(delta_wrt 2).^3)).*(cos(r1).*cos(delta_wrt2)-cos(r2).*cos(del ta_wrt2).^2cos(r2).*sin(delta_wrt2).^2); d_beta_d_omega=((sin(r1).*sin(r2).*sin(omega))./(si n(beta_wrt2).*sin(r3 ).*sin(delta_wrt2).^3)).*(cos(r4).*cos(delta_wrt2)-cos(r3).*sin(delta_wrt2).^2-cos(r3).*cos(delta_wrt2 ).^2); d_phi2_d_omega=d_alpha_d_omega+d_beta_d_omega; %U_bending_J3 = 0.5*K2*(delta_psi2).^2; MJ2 = K2*delta_phi2.*d_phi2_d_omega; M2_1=(MJ2+MJ3); M2_1(145)=(M2_1(144)+M2_1(146))/2; % 1 Square Ansys filename = [ 'ansys_output_1square.txt' ]; string1 = [ 'I:\Thesis\Chester\Thesis Data\Ansys\Combined Data\ ]; fid1 = fopen([string1,filename]); % opens the file ABT = fread(fid1); % reads the file into variable ABT

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76 Appendix B (Continued) fclose(fid1); % closes the data file GBT = native2unicode(ABT)'; % changes data from machine code to text s_iB = findstr( 'my2' ,GBT); % finds end of header A=str2num(GBT(s_iB+4:end)); % turns the data into a numerical matrix roty2_1 = [0; A(:,1)]; % pulls roty2 from file and adds a zero to beginning my2 = [0; A(:,2)]; % pulls my2 from file and adds a zero to beginning my3 = [0; A(:,3)]; % pulls my3 from file and adds a zero to beginning flag2 = abs(sign(my2)); % assigns a zero to values of zero and 1 to values other than zero my_1 = (flag2.*my2+(1-flag2).*my3); %combines my2 and my3 by using previous 1's and zeros to form one column,' my %----------------------------------------------------------------------%2square Analytical L=30; %moment of bending joint2. Ix = a*b^3/12; %second moment of area, compliant mechanisms pg. 399 K2 = -E*Ix./L; %bending coefficient delta_phi2 = (alpha_wrt2(1)-alpha_wrt2)+(beta_wrt2( 1)-beta_wrt2); d_alpha_d_omega=((sin(r1).*sin(omega))./(sin(alpha_ wrt2).*sin(delta_wrt 2).^3)).*(cos(r1).*cos(delta_wrt2)-cos(r2).*cos(del ta_wrt2).^2cos(r2).*sin(delta_wrt2).^2); d_beta_d_omega=((sin(r1).*sin(r2).*sin(omega))./(si n(beta_wrt2).*sin(r3 ).*sin(delta_wrt2).^3)).*(cos(r4).*cos(delta_wrt2)-cos(r3).*sin(delta_wrt2).^2-cos(r3).*cos(delta_wrt2 ).^2); d_phi2_d_omega=d_alpha_d_omega+d_beta_d_omega; %U_bending_J3 = 0.5*K2*(delta_psi2).^2; MJ2 = K2*delta_phi2.*d_phi2_d_omega; M2_2=(MJ2+MJ3); M2_2(145)=(M2_2(144)+M2_2(146))/2; % 2 Square Ansys filename = [ 'ansys_output_2square.txt' ]; string1 = [ 'I:\Thesis\Chester\Thesis Data\Ansys\Combined Data\ ]; fid1 = fopen([string1,filename]); % opens the file ABT = fread(fid1); % reads the file into variable ABT fclose(fid1); % closes the data file GBT = native2unicode(ABT)'; % changes data from machine code to text s_iB = findstr( 'my2' ,GBT); % finds end of header A=str2num(GBT(s_iB+4:end)); % turns the data into a numerical matrix roty2_2 = [0; A(:,1)]; % pulls roty2 from file and adds a zero to beginning

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77 Appendix B (Continued) my2 = [0; A(:,2)]; % pulls my2 from file and adds a zero to beginning my3 = [0; A(:,3)]; % pulls my3 from file and adds a zero to beginning flag2 = abs(sign(my2)); % assigns a zero to values of zero and 1 to values other than zero my_2 = (flag2.*my2+(1-flag2).*my3); %combines my2 and my3 by using previous 1's and zeros to form one column,' my %----------------------------------------------------------------------%3square Analytical L=40; %moment of bending joint2. Ix = a*b^3/12; %second moment of area, compliant mechanisms pg. 399 K2 = -E*Ix./L; %bending coefficient delta_phi2 = (alpha_wrt2(1)-alpha_wrt2)+(beta_wrt2( 1)-beta_wrt2);; d_alpha_d_omega=((sin(r1).*sin(omega))./(sin(alpha_ wrt2).*sin(delta_wrt 2).^3)).*(cos(r1).*cos(delta_wrt2)-cos(r2).*cos(del ta_wrt2).^2cos(r2).*sin(delta_wrt2).^2); d_beta_d_omega=((sin(r1).*sin(r2).*sin(omega))./(si n(beta_wrt2).*sin(r3 ).*sin(delta_wrt2).^3)).*(cos(r4).*cos(delta_wrt2)-cos(r3).*sin(delta_wrt2).^2-cos(r3).*cos(delta_wrt2 ).^2); d_phi2_d_omega=d_alpha_d_omega+d_beta_d_omega; %U_bending_J3 = 0.5*K2*(delta_psi2).^2; MJ2 = K2*delta_phi2.*d_phi2_d_omega; M2_3=(MJ2+MJ3); M2_3(145)=(M2_3(144)+M2_3(146))/2; % 3 Square Ansys filename = [ 'ansys_output_3square.txt' ]; string1 = [ 'I:\Thesis\Chester\Thesis Data\Ansys\Combined Data\ ]; fid1 = fopen([string1,filename]); % opens the file ABT = fread(fid1); % reads the file into variable ABT fclose(fid1); % closes the data file GBT = native2unicode(ABT)'; % changes data from machine code to text s_iB = findstr( 'my2' ,GBT); % finds end of header A=str2num(GBT(s_iB+4:end)); % turns the data into a numerical matrix roty2_3 = [0; A(:,1)]; % pulls roty2 from file and adds a zero to beginning my2 = [0; A(:,2)]; % pulls my2 from file and adds a zero to beginning my3 = [0; A(:,3)]; % pulls my3 from file and adds a zero to beginning flag2 = abs(sign(my2)); % assigns a zero to values of zero and 1 to values other than zero my_3 = (flag2.*my2+(1-flag2).*my3); %combines my2 and my3 by using previous 1's and zeros to form one column,' my

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78 Appendix B (Continued) my_3(156)=0; % deletes incorrect data in 3square program to remo ve spike %----------------------------------------------------------------------%4square Analytical L=50; %moment of bending joint2. Ix = a*b^3/12; %second moment of area, compliant mechanisms pg. 399 K2 = -E*Ix./L; %bending coefficient delta_phi2 = (alpha_wrt2(1)-alpha_wrt2)+(beta_wrt2( 1)-beta_wrt2); d_alpha_d_omega=((sin(r1).*sin(omega))./(sin(alpha_ wrt2).*sin(delta_wrt 2).^3)).*(cos(r1).*cos(delta_wrt2)-cos(r2).*cos(del ta_wrt2).^2cos(r2).*sin(delta_wrt2).^2); d_beta_d_omega=((sin(r1).*sin(r2).*sin(omega))./(si n(beta_wrt2).*sin(r3 ).*sin(delta_wrt2).^3)).*(cos(r4).*cos(delta_wrt2)-cos(r3).*sin(delta_wrt2).^2-cos(r3).*cos(delta_wrt2 ).^2); d_phi2_d_omega=d_alpha_d_omega+d_beta_d_omega; %U_bending_J3 = 0.5*K2*(delta_psi2).^2; MJ2 = K2*delta_phi2.*d_phi2_d_omega; M2_4=(MJ2+MJ3); M2_4(145)=(M2_4(144)+M2_4(146))/2; % 4 Square Ansys filename = [ 'ansys_output_4square.txt' ]; string1 = [ 'I:\Thesis\Chester\Thesis Data\Ansys\Combined Data\ ]; fid1 = fopen([string1,filename]); % opens the file ABT = fread(fid1); % reads the file into variable ABT fclose(fid1); % closes the data file GBT = native2unicode(ABT)'; % changes data from machine code to text s_iB = findstr( 'my2' ,GBT); % finds end of header A=str2num(GBT(s_iB+4:end)); % turns the data into a numerical matrix roty2_4 = [0; A(:,1)]; % pulls roty2 from file and adds a zero to beginning my2 = [0; A(:,2)]; % pulls my2 from file and adds a zero to beginning my3 = [0; A(:,3)]; % pulls my3 from file and adds a zero to beginning flag2 = abs(sign(my2)); % assigns a zero to values of zero and 1 to values other than zero my_4 = (flag2.*my2+(1-flag2).*my3); %combines my2 and my3 by using previous 1's and zeros to form one column,' my my_4(171)=0; % deletes incorrect data in 4square program to remo ve spike %----------------------------------------------------------------------

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79 Appendix B (Continued) %5square Analytical L=60; %moment of bending joint2. Ix = a*b^3/12; %second moment of area, compliant mechanisms pg. 399 K2 = -E*Ix./L; %bending coefficient delta_phi2 = (alpha_wrt2(1)-alpha_wrt2)+(beta_wrt2( 1)-beta_wrt2); d_alpha_d_omega=((sin(r1).*sin(omega))./(sin(alpha_ wrt2).*sin(delta_wrt 2).^3)).*(cos(r1).*cos(delta_wrt2)-cos(r2).*cos(del ta_wrt2).^2cos(r2).*sin(delta_wrt2).^2); d_beta_d_omega=((sin(r1).*sin(r2).*sin(omega))./(si n(beta_wrt2).*sin(r3 ).*sin(delta_wrt2).^3)).*(cos(r4).*cos(delta_wrt2)-cos(r3).*sin(delta_wrt2).^2-cos(r3).*cos(delta_wrt2 ).^2); d_phi2_d_omega=d_alpha_d_omega+d_beta_d_omega; %U_bending_J3 = 0.5*K2*(delta_psi2).^2; MJ2 = K2*delta_phi2.*d_phi2_d_omega; M2_5=(MJ2+MJ3); M2_5(145)=(M2_5(144)+M2_5(146))/2; % 5 Square Ansys filename = [ 'ansys_output_5square.txt' ]; string1 = [ 'I:\Thesis\Chester\Thesis Data\Ansys\Combined Data\ ]; fid1 = fopen([string1,filename]); % opens the file ABT = fread(fid1); % reads the file into variable ABT fclose(fid1); % closes the data file GBT = native2unicode(ABT)'; % changes data from machine code to text s_iB = findstr( 'my2' ,GBT); % finds end of header A=str2num(GBT(s_iB+4:end)); % turns the data into a numerical matrix roty2_5 = [0; A(:,1)]; % pulls roty2 from file and adds a zero to beginning my2 = [0; A(:,2)]; % pulls my2 from file and adds a zero to beginning my3 = [0; A(:,3)]; % pulls my3 from file and adds a zero to beginning flag2 = abs(sign(my2)); % assigns a zero to values of zero and 1 to values other than zero my_5 = (flag2.*my2+(1-flag2).*my3); %combines my2 and my3 by using previous 1's and zeros to form one column,' my my_5(200)=0; % deletes incorrect data in 5square program to remo ve spike %----------------------------------------------------------------------%6square Analytical L=80; %moment of bending joint2.

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80 Appendix B (Continued) Ix = a*b^3/12; %second moment of area, compliant mechanisms pg. 399 K2 = -E*Ix./L; %bending coefficient delta_phi2 = (alpha_wrt2(1)-alpha_wrt2)+(beta_wrt2( 1)-beta_wrt2); d_alpha_d_omega=((sin(r1).*sin(omega))./(sin(alpha_ wrt2).*sin(delta_wrt 2).^3)).*(cos(r1).*cos(delta_wrt2)-cos(r2).*cos(del ta_wrt2).^2cos(r2).*sin(delta_wrt2).^2); d_beta_d_omega=((sin(r1).*sin(r2).*sin(omega))./(si n(beta_wrt2).*sin(r3 ).*sin(delta_wrt2).^3)).*(cos(r4).*cos(delta_wrt2)-cos(r3).*sin(delta_wrt2).^2-cos(r3).*cos(delta_wrt2 ).^2); d_phi2_d_omega=d_alpha_d_omega+d_beta_d_omega; %U_bending_J3 = 0.5*K2*(delta_psi2).^2; MJ2 = K2*delta_phi2.*d_phi2_d_omega; M2_6=(MJ2+MJ3); M2_6(145)=(M2_6(144)+M2_6(146))/2; % 6 Square Ansys filename = [ 'ansys_output_6square.txt' ]; string1 = [ 'I:\Thesis\Chester\Thesis Data\Ansys\Combined Data\ ]; fid1 = fopen([string1,filename]); % opens the file ABT = fread(fid1); % reads the file into variable ABT fclose(fid1); % closes the data file GBT = native2unicode(ABT)'; % changes data from machine code to text s_iB = findstr( 'my2' ,GBT); % finds end of header A=str2num(GBT(s_iB+4:end)); % turns the data into a numerical matrix roty2_6 = [0; A(:,1)]; % pulls roty2 from file and adds a zero to beginning my2 = [0; A(:,2)]; % pulls my2 from file and adds a zero to beginning my3 = [0; A(:,3)]; % pulls my3 from file and adds a zero to beginning flag2 = abs(sign(my2)); % assigns a zero to values of zero and 1 to values other than zero my_6 = (flag2.*my2+(1-flag2).*my3); %combines my2 and my3 by using previous 1's and zeros to form one column,' my my_6(182)=0; % deletes incorrect data in 6square program to remo ve spike %----------------------------------------------------------------------%7square Analytical L=100; %moment of bending joint2. Ix = a*b^3/12; %second moment of area, compliant mechanisms pg. 399 K2 = -E*Ix./L; %bending coefficient delta_phi2 = (alpha_wrt2(1)-alpha_wrt2)+(beta_wrt2( 1)-beta_wrt2);

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81 Appendix B (Continued) d_alpha_d_omega=((sin(r1).*sin(omega))./(sin(alpha_ wrt2).*sin(delta_wrt 2).^3)).*(cos(r1).*cos(delta_wrt2)-cos(r2).*cos(del ta_wrt2).^2cos(r2).*sin(delta_wrt2).^2); d_beta_d_omega=((sin(r1).*sin(r2).*sin(omega))./(si n(beta_wrt2).*sin(r3 ).*sin(delta_wrt2).^3)).*(cos(r4).*cos(delta_wrt2)-cos(r3).*sin(delta_wrt2).^2-cos(r3).*cos(delta_wrt2 ).^2); d_phi2_d_omega=d_alpha_d_omega+d_beta_d_omega; %U_bending_J3 = 0.5*K2*(delta_psi2).^2; MJ2 = K2*delta_phi2.*d_phi2_d_omega; M2_7=(MJ2+MJ3); M2_7(145)=(M2_7(144)+M2_7(146))/2; % 7 Square Ansys filename = [ 'ansys_output_7square.txt' ]; string1 = [ 'I:\Thesis\Chester\Thesis Data\Ansys\Combined Data\ ]; fid1 = fopen([string1,filename]); % opens the file ABT = fread(fid1); % reads the file into variable ABT fclose(fid1); % closes the data file GBT = native2unicode(ABT)'; % changes data from machine code to text s_iB = findstr( 'my2' ,GBT); % finds end of header A=str2num(GBT(s_iB+4:end)); % turns the data into a numerical matrix roty2_7 = [0; A(:,1)]; % pulls roty2 from file and adds a zero to beginning my2 = [0; A(:,2)]; % pulls my2 from file and adds a zero to beginning my3 = [0; A(:,3)]; % pulls my3 from file and adds a zero to beginning flag2 = abs(sign(my2)); % assigns a zero to values of zero and 1 to values other than zero my_7 = (flag2.*my2+(1-flag2).*my3); %combines my2 and my3 by using previous 1's and zeros to form one column,' my my_7(168)=0; % deletes incorrect data in 7square program to remo ve spike %----------------------------------------------------------------------%8square Analytical L=120; %moment of bending joint2. Ix = a*b^3/12; %second moment of area, compliant mechanisms pg. 399 K2 = -E*Ix./L; %bending coefficient delta_phi2 = (alpha_wrt2(1)-alpha_wrt2)+(beta_wrt2( 1)-beta_wrt2); d_alpha_d_omega=((sin(r1).*sin(omega))./(sin(alpha_ wrt2).*sin(delta_wrt 2).^3)).*(cos(r1).*cos(delta_wrt2)-cos(r2).*cos(del ta_wrt2).^2cos(r2).*sin(delta_wrt2).^2); d_beta_d_omega=((sin(r1).*sin(r2).*sin(omega))./(si n(beta_wrt2).*sin(r3 ).*sin(delta_wrt2).^3)).*(cos(r4).*cos(delta_wrt2)-cos(r3).*sin(delta_wrt2).^2-cos(r3).*cos(delta_wrt2 ).^2);

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82 Appendix B (Continued) d_phi2_d_omega=d_alpha_d_omega+d_beta_d_omega; %U_bending_J3 = 0.5*K2*(delta_psi2).^2; MJ2 = K2*delta_phi2.*d_phi2_d_omega; M2_8=(MJ2+MJ3); M2_8(145)=(M2_8(144)+M2_8(146))/2; % 8 Square Ansys filename = [ 'ansys_output_8square.txt' ]; string1 = [ 'I:\Thesis\Chester\Thesis Data\Ansys\Combined Data\ ]; fid1 = fopen([string1,filename]); % opens the file ABT = fread(fid1); % reads the file into variable ABT fclose(fid1); % closes the data file GBT = native2unicode(ABT)'; % changes data from machine code to text s_iB = findstr( 'my2' ,GBT); % finds end of header A=str2num(GBT(s_iB+4:end)); % turns the data into a numerical matrix roty2_8 = [0; A(:,1)]; % pulls roty2 from file and adds a zero to beginning my2 = [0; A(:,2)]; % pulls my2 from file and adds a zero to beginning my3 = [0; A(:,3)]; % pulls my3 from file and adds a zero to beginning flag2 = abs(sign(my2)); % assigns a zero to values of zero and 1 to values other than zero my_8 = (flag2.*my2+(1-flag2).*my3); %combines my2 and my3 by using previous 1's and zeros to form one column,' my my_8(177)=0; % deletes incorrect data in 8square program to remo ve spike %----------------------------------------------------------------------%individual plots of ansys vs analytical figure(1) clf hold on set(gca, 'fontsize' ,15) xlabel( '\theta_2 (Degrees)' 'fontsize' ,17) ylabel( 'M_2 (\muN\mum)' 'fontsize' ,17) h=[plot(roty2_1*180/pi,my_1, 'k' ) plot(omega_match_ansys*180/pi,M2_1, 'b' )]; p=legend(h, 'FEA' 'Pure Bending' ); set(p, 'box' 'off' ) set(p, 'location' 'northwest' ) set(p, 'fontsize' ,15) print(gcf, '-dtiff' '-r300' 'Graph_MomVsRot_PureBending_1' ) figure(2) clf hold on

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83 Appendix B (Continued) set(gca, 'fontsize' ,15) xlabel( '\theta_2 (Degrees)' 'fontsize' ,17) ylabel( 'M_2 (\muN\mum)' 'fontsize' ,17) h=[plot(roty2_2*180/pi,my_2, 'k' ) plot(omega_match_ansys*180/pi,M2_2, 'b' )]; p=legend(h, 'FEA' 'Pure Bending' ); set(p, 'box' 'off' ) set(p, 'location' 'northwest' ) set(p, 'fontsize' ,15) print(gcf, '-dtiff' '-r300' 'Graph_MomVsRot_PureBending_2' ) figure(3) clf hold on set(gca, 'fontsize' ,15) xlabel( '\theta_2 (Degrees)' 'fontsize' ,17) ylabel( 'M_2 (\muN\mum)' 'fontsize' ,17) h=[plot(roty2_3*180/pi,my_3, 'k' ) plot(omega_match_ansys*180/pi,M2_3, 'b' )]; p=legend(h, 'FEA' 'Pure Bending' ); set(p, 'box' 'off' ) set(p, 'location' 'northwest' ) set(p, 'fontsize' ,15) print(gcf, '-dtiff' '-r300' 'Graph_MomVsRot_PureBending_3' ) figure(4) clf hold on set(gca, 'fontsize' ,15) xlabel( '\theta_2 (Degrees)' 'fontsize' ,17) ylabel( 'M_2 (\muN\mum)' 'fontsize' ,17) h=[plot(roty2_4*180/pi,my_4, 'k' ) plot(omega_match_ansys*180/pi,M2_4, 'b' )]; p=legend(h, 'FEA' 'Pure Bending' ); set(p, 'box' 'off' ) set(p, 'location' 'northwest' ) set(p, 'fontsize' ,15) print(gcf, '-dtiff' '-r300' 'Graph_MomVsRot_PureBending_4' ) figure(5) clf hold on set(gca, 'fontsize' ,15) xlabel( '\theta_2 (Degrees)' 'fontsize' ,17) ylabel( 'M_2 (\muN\mum)' 'fontsize' ,17) h=[plot(roty2_5*180/pi,my_5, 'k' ) plot(omega_match_ansys*180/pi,M2_5, 'b' )]; p=legend(h, 'FEA' 'Pure Bending' ); set(p, 'box' 'off' ) set(p, 'location' 'northwest' ) set(p, 'fontsize' ,15) print(gcf, '-dtiff' '-r300' 'Graph_MomVsRot_PureBending_5' ) figure(6)

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84 Appendix B (Continued) clf hold on set(gca, 'fontsize' ,15) xlabel( '\theta_2 (Degrees)' 'fontsize' ,17) ylabel( 'M_2 (\muN\mum)' 'fontsize' ,17) h=[plot(roty2_6*180/pi,my_6, 'k' ) plot(omega_match_ansys*180/pi,M2_6, 'b' )]; p=legend(h, 'FEA' 'Pure Bending' ); set(p, 'box' 'off' ) set(p, 'location' 'northwest' ) set(p, 'fontsize' ,15) print(gcf, '-dtiff' '-r300' 'Graph_MomVsRot_PureBending_6' ) figure(7) clf hold on set(gca, 'fontsize' ,15) xlabel( '\theta_2 (Degrees)' 'fontsize' ,17) ylabel( 'M_2 (\muN\mum)' 'fontsize' ,17) h=[plot(roty2_7*180/pi,my_7, 'k' ) plot(omega_match_ansys*180/pi,M2_7, 'b' )]; p=legend(h, 'FEA' 'Pure Bending' ); set(p, 'box' 'off' ) set(p, 'location' 'northwest' ) set(p, 'fontsize' ,15) print(gcf, '-dtiff' '-r300' 'Graph_MomVsRot_PureBending_7' ) figure(8) clf hold on set(gca, 'fontsize' ,15) xlabel( '\theta_2 (Degrees)' 'fontsize' ,17) ylabel( 'M_2 (\muN\mum)' 'fontsize' ,17) h=[plot(roty2_8*180/pi,my_8, 'k' ) plot(omega_match_ansys*180/pi,M2_8, 'b' )]; p=legend(h, 'FEA' 'Pure Bending' ); set(p, 'box' 'off' ) set(p, 'location' 'northwest' ) set(p, 'fontsize' ,15) print(gcf, '-dtiff' '-r300' 'Graph_MomVsRot_PureBending_8' ) %----------------------------------------------------------------------%pure bending combined plots figure(9) clf hold on set(gca, 'fontsize' ,15) xlabel( '\theta_2 (Degrees)' 'fontsize' ,17) ylabel( 'M_2 (\muN\mum)' 'fontsize' ,17) h=[plot(omega_match_ansys(1)*180/pi,M2_1(1), '*' )

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85 Appendix B (Continued) plot(omega_match_ansys*180/pi,M2_1, 'k' ) plot(omega_match_ansys*180/pi,M2_2, 'r' ) plot(omega_match_ansys*180/pi,M2_3, 'm' ) plot(omega_match_ansys*180/pi,M2_4, 'b' ) plot(omega_match_ansys*180/pi,M2_5, 'g' ) plot(omega_match_ansys*180/pi,M2_6, 'c' ) plot(omega_match_ansys*180/pi,M2_7, 'k' ) plot(omega_match_ansys*180/pi,M2_8, 'r' )]; p=legend(h, 'Start' 'Model 1' 'Model 2' 'Model 3' 'Model 4' 'Model 5' 'Model 6' 'Model 7' 'Model 8' ); set(p, 'box' 'off' ) set(p, 'location' 'northeast' ) set(p, 'fontsize' ,11) print(gcf, '-dtiff' '-r300' 'Graph_MomVsRot_PureBending_All' ) B.3 Combined FEA Plots The following program plots the FEA results for each mo del. clc % 1 Square filename = [ 'ansys_output_1square.txt' ]; string1 = [ 'I:\Thesis\Chester\Thesis Data\Ansys\Combined Data\ ]; fid1 = fopen([string1,filename]); % opens the file ABT = fread(fid1); % reads the file into variable ABT fclose(fid1); % closes the data file GBT = native2unicode(ABT)'; % changes data from machine code to text s_iB = findstr( 'my2' ,GBT); % finds end of header A=str2num(GBT(s_iB+4:end)); % turns the data into a numerical matrix roty2_1 = [0; A(:,1)]; % pulls roty2 from file and adds a zero to beginning my2 = [0; A(:,2)]; % pulls my2 from file and adds a zero to beginning my3 = [0; A(:,3)]; % pulls my3 from file and adds a zero to beginning flag2 = abs(sign(my2)); % assigns a zero to values of zero and 1 to values other than zero my_1 = (flag2.*my2+(1-flag2).*my3); %combines my2 and my3 by using previous 1's and zeros to form one column,' my %----------------------------------------------------------------------% 2 Square filename = [ 'ansys_output_2square.txt' ]; string1 = [ 'I:\Thesis\Chester\Thesis Data\Ansys\Combined Data\ ]; fid1 = fopen([string1,filename]); % opens the file ABT = fread(fid1); % reads the file into variable ABT

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86 Appendix B (Continued) fclose(fid1); % closes the data file GBT = native2unicode(ABT)'; % changes data from machine code to text s_iB = findstr( 'my2' ,GBT); % finds end of header A=str2num(GBT(s_iB+4:end)); % turns the data into a numerical matrix roty2_2 = [0; A(:,1)]; % pulls roty2 from file and adds a zero to beginning my2 = [0; A(:,2)]; % pulls my2 from file and adds a zero to beginning my3 = [0; A(:,3)]; % pulls my3 from file and adds a zero to beginning flag2 = abs(sign(my2)); % assigns a zero to values of zero and 1 to values other than zero my_2 = (flag2.*my2+(1-flag2).*my3); %combines my2 and my3 by using previous 1's and zeros to form one column,' my %----------------------------------------------------------------------% 3 Square filename = [ 'ansys_output_3square.txt' ]; string1 = [ 'I:\Thesis\Chester\Thesis Data\Ansys\Combined Data\ ]; fid1 = fopen([string1,filename]); % opens the file ABT = fread(fid1); % reads the file into variable ABT fclose(fid1); % closes the data file GBT = native2unicode(ABT)'; % changes data from machine code to text s_iB = findstr( 'my2' ,GBT); % finds end of header A=str2num(GBT(s_iB+4:end)); % turns the data into a numerical matrix roty2_3 = [0; A(:,1)]; % pulls roty2 from file and adds a zero to beginning my2 = [0; A(:,2)]; % pulls my2 from file and adds a zero to beginning my3 = [0; A(:,3)]; % pulls my3 from file and adds a zero to beginning flag2 = abs(sign(my2)); % assigns a zero to values of zero and 1 to values other than zero my_3 = (flag2.*my2+(1-flag2).*my3); %combines my2 and my3 by using previous 1's and zeros to form one column,' my %----------------------------------------------------------------------% 4 Square filename = [ 'ansys_output_4square.txt' ]; string1 = [ 'I:\Thesis\Chester\Thesis Data\Ansys\Combined Data\ ]; fid1 = fopen([string1,filename]); % opens the file ABT = fread(fid1); % reads the file into variable ABT fclose(fid1); % closes the data file GBT = native2unicode(ABT)'; % changes data from machine code to text s_iB = findstr( 'my2' ,GBT); % finds end of header

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87 Appendix B (Continued) A=str2num(GBT(s_iB+4:end)); % turns the data into a numerical matrix roty2_4 = [0; A(:,1)]; % pulls roty2 from file and adds a zero to beginning my2 = [0; A(:,2)]; % pulls my2 from file and adds a zero to beginning my3 = [0; A(:,3)]; % pulls my3 from file and adds a zero to beginning flag2 = abs(sign(my2)); % assigns a zero to values of zero and 1 to values other than zero my_4 = (flag2.*my2+(1-flag2).*my3); %combines my2 and my3 by using previous 1's and zeros to form one column,' my %----------------------------------------------------------------------% 5 Square filename = [ 'ansys_output_5square.txt' ]; string1 = [ 'I:\Thesis\Chester\Thesis Data\Ansys\Combined Data\ ]; fid1 = fopen([string1,filename]); % opens the file ABT = fread(fid1); % reads the file into variable ABT fclose(fid1); % closes the data file GBT = native2unicode(ABT)'; % changes data from machine code to text s_iB = findstr( 'my2' ,GBT); % finds end of header A=str2num(GBT(s_iB+4:end)); % turns the data into a numerical matrix roty2_5 = [0; A(:,1)]; % pulls roty2 from file and adds a zero to beginning my2 = [0; A(:,2)]; % pulls my2 from file and adds a zero to beginning my3 = [0; A(:,3)]; % pulls my3 from file and adds a zero to beginning flag2 = abs(sign(my2)); % assigns a zero to values of zero and 1 to values other than zero my_5 = (flag2.*my2+(1-flag2).*my3); %combines my2 and my3 by using previous 1's and zeros to form one column,' my %----------------------------------------------------------------------% 6 Square filename = [ 'ansys_output_6square.txt' ]; string1 = [ 'I:\Thesis\Chester\Thesis Data\Ansys\Combined Data\ ]; fid1 = fopen([string1,filename]); % opens the file ABT = fread(fid1); % reads the file into variable ABT fclose(fid1); % closes the data file GBT = native2unicode(ABT)'; % changes data from machine code to text s_iB = findstr( 'my2' ,GBT); % finds end of header A=str2num(GBT(s_iB+4:end)); % turns the data into a numerical matrix roty2_6 = [0; A(:,1)]; % pulls roty2 from file and adds a zero to beginning

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88 Appendix B (Continued) my2 = [0; A(:,2)]; % pulls my2 from file and adds a zero to beginning my3 = [0; A(:,3)]; % pulls my3 from file and adds a zero to beginning flag2 = abs(sign(my2)); % assigns a zero to values of zero and 1 to values other than zero my_6 = (flag2.*my2+(1-flag2).*my3); %combines my2 and my3 by using previous 1's and zeros to form one column,' my %----------------------------------------------------------------------% 7 Square filename = [ 'ansys_output_7square.txt' ]; string1 = [ 'I:\Thesis\Chester\Thesis Data\Ansys\Combined Data\ ]; fid1 = fopen([string1,filename]); % opens the file ABT = fread(fid1); % reads the file into variable ABT fclose(fid1); % closes the data file GBT = native2unicode(ABT)'; % changes data from machine code to text s_iB = findstr( 'my2' ,GBT); % finds end of header A=str2num(GBT(s_iB+4:end)); % turns the data into a numerical matrix roty2_7 = [0; A(:,1)]; % pulls roty2 from file and adds a zero to beginning my2 = [0; A(:,2)]; % pulls my2 from file and adds a zero to beginning my3 = [0; A(:,3)]; % pulls my3 from file and adds a zero to beginning flag2 = abs(sign(my2)); % assigns a zero to values of zero and 1 to values other than zero my_7 = (flag2.*my2+(1-flag2).*my3); %combines my2 and my3 by using previous 1's and zeros to form one column,' my %----------------------------------------------------------------------% 8 Square filename = [ 'ansys_output_8square.txt' ]; string1 = [ 'I:\Thesis\Chester\Thesis Data\Ansys\Combined Data\ ]; fid1 = fopen([string1,filename]); % opens the file ABT = fread(fid1); % reads the file into variable ABT fclose(fid1); % closes the data file GBT = native2unicode(ABT)'; % changes data from machine code to text s_iB = findstr( 'my2' ,GBT); % finds end of header A=str2num(GBT(s_iB+4:end)); % turns the data into a numerical matrix roty2_8 = [0; A(:,1)]; % pulls roty2 from file and adds a zero to beginning my2 = [0; A(:,2)]; % pulls my2 from file and adds a zero to beginning my3 = [0; A(:,3)]; % pulls my3 from file and adds a zero to beginning

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89 Appendix B (Continued) flag2 = abs(sign(my2)); % assigns a zero to values of zero and 1 to values other than zero my_8 = (flag2.*my2+(1-flag2).*my3); %combines my2 and my3 by using previous 1's and zeros to form one column,' my %----------------------------------------------------------------------my_3(156)=0; % deletes incorrect data in 3square program to remo ve spike my_4(171)=0; % deletes incorrect data in 4square program to remo ve spike my_5(200)=0; % deletes incorrect data in 5square program to remo ve spike my_6(182)=0; % deletes incorrect data in 6square program to remo ve spike my_7(168)=0; % deletes incorrect data in 7square program to remo ve spike my_8(177)=0; % deletes incorrect data in 8square program to remo ve spike %for the sake of the paper, 2square has been re-lab eled model 1. figure(1) clf hold on set(gca, 'fontsize' ,15) xlabel( '\theta_2 (Degrees)' 'fontsize' ,17) ylabel( 'M_2 (\muN\mum)' 'fontsize' ,17) h=[plot(roty2_1(1),my_1(1), 'b*' ) plot(roty2_2*180/pi,my_2, 'k' ) plot(roty2_3*180/pi,my_3, 'r' ) plot(roty2_4*180/pi,my_4, 'm' ) plot(roty2_5*180/pi,my_5, 'b' ) plot(roty2_6*180/pi,my_6, 'g' ) plot(roty2_7*180/pi,my_7, 'c' ) plot(roty2_8*180/pi,my_8, 'k' )]; p=legend(h, 'Start' 'model 1' 'model 2' 'model 3' 'model 4' 'model 5' 'model 6' 'model 7' ); set(p, 'box' 'off' ) set(p, 'location' 'northwest' ) set(p, 'fontsize' ,15) print(gcf, '-dtiff' '-r300' 'Graph_MomVsRot_Ansys_Combined_2-8' ) %----------------------------------------------------------------------figure(2) clf hold on set(gca, 'fontsize' ,15) xlabel( '\theta_2 (Degrees)' 'fontsize' ,17) ylabel( 'M_2 (\muN\mum)' 'fontsize' ,17) h=[plot(roty2_1(1),my_1(1), 'b*' ) plot(roty2_1*180/pi,my_1, 'k' ) plot(roty2_2*180/pi,my_2, 'r' ) plot(roty2_3*180/pi,my_3, 'm' )

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90 Appendix B (Continued) plot(roty2_4*180/pi,my_4, 'b' ) plot(roty2_5*180/pi,my_5, 'g' ) plot(roty2_6*180/pi,my_6, 'c' ) plot(roty2_7*180/pi,my_7, 'k' ) plot(roty2_8*180/pi,my_8, 'r' )]; p=legend(h, 'Start' 'model 1' 'model 2' 'model 3' 'model 4' 'model 5' 'model 6' 'model 7' 'model 8' ); set(p, 'box' 'off' ) set(p, 'location' 'northeast' ) set(p, 'fontsize' ,15) print(gcf, '-dtiff' '-r300' 'Graph_MomVsRot_Ansys_Combined_1-8' ) %----------------------------------------------------------------------


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Smith, Chester
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Modeling and parameter study of bistable spherical compliant mechanisms
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by Chester Smith.
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[Tampa, Fla] :
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2011.
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Thesis
(M.S.M.E.)--University of South Florida, 2011.
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Text (Electronic thesis) in PDF format.
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ABSTRACT: The bistable spherical compliant mechanism (BSCM) is a novel device capable of large, repeatable, out-of-plane motion, characteristics that are somewhat difficult to achieve with surface micromachined microelectromechanical systems. An improved pseudo-rigid-body model (PRBM) to predict the behavior of the BSCM is presented. The new model was used to analyze seven different versions of the device, each with a different compliant joint length. The new model, which adds torsion, is compared with a finite element analysis (FEA) beam model. The new model more closely approximates the results yielded by FEA than previous models used to analyze the BSCM. Future work is needed to quantify stress-stiffening interactions between bending and torsion. Both FEA and the current models show that increasing the length of the compliant segment decreases the amount of force required to actuate the device.
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Advisor:
Lusk, Craig .
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x Engineering Mechanical Engineering
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