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Kinematics of curved flexible beam

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Title:
Kinematics of curved flexible beam
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English
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Jagirdar, Saurabh
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University of South Florida
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Tampa, Fla
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Subjects / Keywords:
Compliant mechanisms
Pseudo-rigid-body
Spherical
MEMS
Out of plane
Dissertations, Academic -- Mechanical Engineering -- Masters -- USF   ( lcsh )
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bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
ABSTRACT: Compliant mechanism theory permits a procedure called rigidbody replacement, in which two or more rigid links of the mechanism are replaced by a compliant flexure with equivalent motion. Methods for designing flexure with equivalent motion to replace rigid links are detailed in Pseudo-Rigid-Body Models (PRBMs). Such models have previously been developed for planar mechanisms. This thesis develops the first PRBM for spherical mechanisms. In formulating this PRBM for a spherical mechanism, we begin by applying displacements are applied to a curved beam that cause it todeflect in a manner consistent with spherical kinematics. The motion of the beam is calculated using Finite Element Analysis. These results areanalyzed to give the PRBM parameters. These PRBM parameters vary with the arc length and the aspect ratio of the curved beam.
Thesis:
Thesis (M.S.M.E.)--University of South Florida, 2006.
Bibliography:
Includes bibliographical references.
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Mode of access: World Wide Web.
Statement of Responsibility:
by Saurabh Jagirdar.
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Title from PDF of title page.
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Document formatted into pages; contains 120 pages.

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oclc - 190654786
usfldc doi - E14-SFE0001853
usfldc handle - e14.1853
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ABSTRACT: Compliant mechanism theory permits a procedure called rigidbody replacement, in which two or more rigid links of the mechanism are replaced by a compliant flexure with equivalent motion. Methods for designing flexure with equivalent motion to replace rigid links are detailed in Pseudo-Rigid-Body Models (PRBMs). Such models have previously been developed for planar mechanisms. This thesis develops the first PRBM for spherical mechanisms. In formulating this PRBM for a spherical mechanism, we begin by applying displacements are applied to a curved beam that cause it todeflect in a manner consistent with spherical kinematics. The motion of the beam is calculated using Finite Element Analysis. These results areanalyzed to give the PRBM parameters. These PRBM parameters vary with the arc length and the aspect ratio of the curved beam.
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Kinematics of Curved Flexible Beam by Saurabh Jagirdar A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Department of Mechanical Engineering College of Engineering University of South Florida Major Professor: Craig P. Lusk, Ph.D. Rajiv Dubey, Ph.D. Autar K. Kaw, Ph.D. Date of Approval: October 26, 2006 Keywords: Compliant mechanisms, Pseudo-rigid-body, Spherical, MEMS, Out of plane Copyright 2006, Saurabh Jagirdar

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Dedication Dedicated to my parents and my major professor

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Acknowledgement I wish to express my gratitude to everyone who contributed to making this thesis a reality. I must single out my professor Dr. Craig P. Lusk who supported and guided me right from the beginning to bring this thesis to fruition. I also want to thank my supe rvisory committee Dr. Rajiv Dubey and Dr Autar K. Kaw and all other professors for their encouragement and guidance. I am especially grateful to our department staff Ms Susan Britten, Ms Shirley Tervort and Mr. Wes Frusher who helped me through all the official procedur es and setting up our compliant mechanisms laboratory. I thank Ms Cherine Chehab from the College of Engineering, USF to edit and improve the format of the thesis. I am indebted to Mr Prateek Asth ana of CSEE, Dept, USF to help me generate large number of input fi les by writing just one program. This program saved me enormous amo unt of time that it would have taken to generate them one by one. I thank my friends and colleagues of the Mechanical engineering department and other departments for making my life fun and also

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helping me through various ways, a sp ecial reference to Daniel Vilceus who continuously kept me pepped up with his sense of humour, Hari Patel, John Daly, Shantanu Shevad e, Aditya Bansal, Cesar Hernandez and Son Ho. I also thank my laboratory mates Joe, Alex, Diego, Sebastian, Patricia and Issa for their help and support. I am deeply indebted to my roommates Dr Apurva Panchal and Phaninder Injeti for their patien ce to bear and take all my eccentricities and help me through my tough times. I once again thank the Mechanical Engineering Department., the College of Engineering and University of South Florida, Tampa Florida.

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i Table of Contents List of Tables ii List of Figures iii Abstract vi 1. Introduction 1 1.1 Scope 2 1.2 Background 4 1.3 Roadmap 15 2. Methodology and Model Development 16 2.1 Correspondence between spherical and planar PRBMs 16 2.2 Kinematics of compliant circular arc 18 2.3 Spherical kinematics of the pseudo-rigid-body model 21 2.4 Spherical loading condition analogous to planar vertical end load 23 3. Finite Element Analysis (FEA) 24 4. Parametric Approximation of the Curved Beams Deflection Path 33 5. Results and Discussion 40 6. Conclusion 50 References 51 Appendices 56 Appendix A Spherical Triangles and Napier Rules 57 Appendix B Manual for FEA 62 Appendix C Algorithms to Find 100 Appendix D Summary 119

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ii List of Tables Table 1: Spherical PRBM 119 Table 2: Planar PRBM 120

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iii List of Figures Figure 1: A PRBM for a cantilever beam with a vertical end load (Howell 2001) 7 Figure 2: Planar Mechanism with sliders moving on perpendicular straight lines 9 Figure 3: Spherical mechanism with sliders moving on perpendicular circular arcs 10 Figure 4: Geodesics 11 Figure 5: Parallel transport (Henderson 1998) 13 Figure 6: Parallel transport along the same longitude 14 Figure 7: Relationship between existing planar PRBM and the spherical PRBM developed in this work 17 Figure 8: Reference frames describing the motion of the end of a compliant circular cantilever 20 Figure 9: The pseudo-rigid-bod y model of the compliant curved beam 22 Figure 10: Path followed by beam the dotted line from Q to Q’’ 25 Figure 11: Reference frames used to model the spherical mechanism and its planar equivalent 26 Figure 12: Cross-section of beam for various aspect ratios 29 Figure 13: Finite element model 30 Figure 14 : Deflection of curved segment 35 Figure 15: Deflection of PRBM 37

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iv Figure 16: Final position of beam from fixed end v/s input displacement 41 Figure 17: Deflection of beam about neutral axis, 0,v/s input displacement 41 Figure 18: v/s Arc-lengths showing various colors for aspect ratios 42 Figure 19: C v/s Arc-lengths showing variou s colors for aspect ratios43 Figure 20: max v/s Arc-lengths showing va rious colors for aspect ratios 43 Figure 21: v/s Arc-lengths, for 200 load-steps of input displacement 45 Figure 22: C v/s Arc-lengths, for 200 load-steps of input displacement 46 Figure 23: max v/s Arc-lengths, for 200 load-steps of input displacement 47 Figure 24: Trend-line of for aspect ratio 0.1 48 Figure 25: Trend-line of for aspect ratio 0.4 48 Figure 26: Trend-line of for aspect ratio 0.7 49 Figure 27: Spherical triangles 57 Figure 28: Five parts arrang ed in order of occurrence 58 Figure 29: Spherical right triangle 59 Figure 30: Five parts for PRBM right spherical triangle 60 Figure 31: Activating Graphical User Interface (GUI) 62 Figure 32: Limiting the GUI opti ons to structural preferences 63 Figure 33: Adding or defining new element types 64

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v Figure 34: Beam elements 65 Figure 35: Defining real consta nts for respective elements 66 Figure 36: Inputting area and moment of inertia values to elements 67 Figure 37: Defining material properties 68 Figure 38: Creating key-points on work-plane through GUI 69 Figure 39: Creating key-po ints using command line 70 Figure 40: Pan, zoom, rotate 71 Figure 41: Defining of orthogonal triad at beam end 72 Figure 42: Creating lines 73 Figure 43: Creating arcs 74 Figure 44: Meshing 75 Figure 45: Mesh attributes for line 76 Figure 46: Allocating specific material to mesh (elements) 77 Figure 47: Selecting analysis type 78 Figure 48: Large displacement analysis selected 79 Figure 49: Equation chosen solvers 80 Figure 50: Applying loads to the beam 81 Figure 51: Solve 82 Figure 52: During solve 83 Figure 53: Output 84 Figure 54: To get the log-file 85

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vi Kinematics of Curved Flexible Beam Saurabh Jagirdar ABSTRACT Compliant mechanism theory permits a procedure called rigidbody replacement, in which two or more rigid links of the mechanism are replaced by a compliant flexur e with equivalent motion. Methods for designing flexure with equivalent motion to replace rigid links are detailed in Pseudo-Rigid-Body Mode ls (PRBMs). Such models have previously been developed for pl anar mechanisms. This thesis develops the first PRBM fo r spherical mechanisms. In formulating this PRBM for a sp herical mechanism, we begin by applying displacements are applied to a curved beam that cause it to deflect in a manner consistent with spherical kinematics. The motion of the beam is calculated using Finite Element Analysis. These results are analyzed to give the PRBM parameters. These PRBM parameters vary with the arc length and the as pect ratio of the curved beam.

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1 1. Introduction Mechanisms have been defined as “mechanical devices for transferring motion and/or force fr om a source to an output” (Erdman et al. 2001). Mechanisms form an important part of how our modern society interacts with the world, whether it is the steering wheel, the computer keyboard, or even the ha ndle of a door. Most mechanisms are systems of levers, cams and gears, which move and rotate, and which have rigid parts. Compliant mechanisms are mechanisms that “gain some or all of their ability to move from the deflection of flexible segments” (Salamon 1989). In complia nt mechanisms, individual parts not only move and rotate, but also undergo elastic deformations in response to the forces which ar e imposed on them. Some common compliant mechanisms are binder clip s, paper clips, backpack latch, lid, nail-clippers, etc. Compliant mechanisms can have improved performance, lower costs and greate r potential functional integration when compared with rigid-body mechanisms (Her 1986, Sevak and McLarman 1974).

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2 1.1 Scope Compliant mechanism theory permits a procedure called rigidbody replacement, in which two or more rigid links of the mechanism are replaced by a compliant flexur e with equivalent motion (Howell 2001). Methods for designing flexure wi th equivalent motion to replace rigid links are detailed in Pseudo-Rig id-Body Models (PRBMs). In many texts, (Boettama and Roth, Mc Cart hy 2000), rigid body analysis of synthesis techniques have been cl assified as planar, spherical and spatial according to the type of vector algebra used to describe the mechanisms. In a planar mechanism, the path of any single part of a link lies in a plane and in a spherical mechanism, the path of any single part of a link lies on the surface of a sphere. Numerous PRBMs have been deve loped for planar mechanisms by Midha et al (1992, 2000), Ho well and Midha (1994a, 1994b, 1995) Saxena and Kramer (1998), and Dado (2001) and used in applications such as Microelectromechanical Sy stems (MEMS) (Baker et al. 2000, Hubbard 2005, Ananthasuresh et al 1993, Ananthasuresh and Kota 1996, Jensen et al. 1997, Salmon et al. 1996 and Kota et al. 2001), prosthetics (Guerinot et al. 2004), cl utches (Roach et al. 1998, Crane et al. 2004), micro-bearings (Ca nnon et al. 2005), constant-force mechanisms (Millar et al. 1996), parallel mechanisms (Derderian et al. 1996), and bi-stable mechanisms (J ensen et al. 1999) and used in

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3 various other applications like th ermal and electrical actuating mechanisms for MEMS (Brocket an d Stokes (1991) and Saggere and Kota (1997)). Thus, extensive rese arch has been done on planar compliant mechanisms using PRBMs. A prime advantage of compliant mechanisms is the part count reduction, that is, flexures can replace rigid links and reduce the number of joints (Howell 2001). This plays a significant role in the fabrication of MEMS. In MEMS desi gn, the increase in the number of joints directly increases the comple xity to manufacture MEMS. (Howell 2001). Compliant mechanisms also have increased precision, increased reliability, reduced weight and re duced maintenance (Howell 2001). These advantages make compliant me chanisms ideal for MEMS design and hence the applications for MEMS using compliant mechanisms are abundant. The PRBM concept has been particul arly fruitful in the design of surface micro-machined MEMS. Su rface micromachining is less expensive and more versatile than alternative forms of fabrication (Howell 2001). For these reasons much of current MEMS research is devoted to this technique. But MEMS designs, fabricated by surface micro-machining are limited to moving back-and-forth and side-to-side (two dimensional motion) i.e. surface micro-machined devices are essentially flat (or in-plane or planar). For applications that need a

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4 micro mechanism that rotates out of the plane of fabrication with an in-plane rotational input, or that rotates spatially about a point, existing planar compliant mechanisms are not suitable. Given that all current PRBMs relate compliant mechanisms to planar rigid-body mechanisms, we are led to ask is it possible to derive PRBMs that relate compliant mechanisms to sp herical rigid-body mechanisms. No such PRBMs have been developed for spherical mechanisms. It is anticipated that the description of compliant spherical mechanisms with spherical motion will simplify the design of MEMS with out of plane motion. In this thesis, the first PR BM for a spherical compliant mechanism is developed. The kinema tics of a curved flexure with the equivalent of a vertical end load is studied and a spherical PRBM for a curved cantilever beam is developed by approximating the motion of the compliant flexure as an equi valent rigid-body mechanism. 1.2 Background The motion of rigid-body mech anisms can be analyzed with matrix algebra (McCarthy 2000) or othe r techniques and more sophisticated techniques are requir ed for spherical mechanisms than planar mechanisms. The analysis of the motion of compliant mechanisms, on the other hand, usually requires the solution of

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5 differential equations, which describe the physics of an infinitely thin section of the mechanism (Frisch Fa y 1962). Because the terms planar and spherical describe the gross motion of objects of finite size, it is not obvious a priori when or if these terms apply to compliant mechanisms. However, a compliant mechanism may be termed as planar or spherical mechanism when the solution of its governing differential equations can be reasonably approximated with rigid-body mathematical techniques i.e. matrix algebra. To conv ert the solution method of a compliant mechanis m from a differential equation approach to an algebraic approa ch, a number of assumptions and specifications need to be made The differential equation gives information about the rela tionships of a continuous series of points in the mechanism; the algebraic equati on gives information about a few specific points. Thus, the transition requires the specification of the points of interest, typically the en ds of the flexible segment. The solution to the differential equation s requires that boundary conditions, i.e. information about applied load s and displacements, be specified (Howell 2001). Thus, the conversion to an algebraic solution is valid only for the specific loading condit ions. These restrictions usually are placed on loading directions rath er than magnitudes (Howell 2001). A validated and accurate identification between a spherical compliant

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6 mechanism and a rigid-body mechanis m with equivalent motion at the points of interest is a spherical PRBM. The PRBM consists of diagrams and equations describing the flexible member and gives a rigidlink equivalent of the compliant mechanism which has the same mo tion and flexibility for a known range of motion and to a known ma thematical tolerance. A PRBM can be used to perform analysis (i.e. given a compliant flexure, its motion can be found by treating it as th e rigid body) or design (given a particular desired motion, a rigid body mechanism that performs the motion can be found, and the PRBM can be used to convert that rigidbody mechanism into a compliant me chanism). The creation of a PRBM entails steps beyond the typical mathem atical analysis of motion of the compliant segment. These additional steps are necessary to find a simple and accurate rigid-body a pproximation of the motion of the compliant segment. Once that rigid-body approximation has been identified, it is optimized and validat ed so that its range of applicability and level of error is known and acceptable. This identification step requires proposing a topology for the rigid-body mechanism, i.e. specification of the number of links and joints. The optimization and validation of steps involve using a nu merical optimization routine that insures that the rigid body approx imation has a tolerable error (less than 0.5%) over as large a range of motion as possible. The creation

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7 of such PRBMs is justified because th ey are easy to use in design and because the use of the PRBM in conn ection with rigid-body synthesis techniques produces compliant me chanism configurations that are unlikely to be produced in any other way. An example of this approach is the PRBM for a straight cantilever beam with vertical end load (Howell 2001), which associates motion of a compliant flexure with a rigid-link mechanism as shown in Figure 1. Figure 1(a) shows a straight cantilever beam subjec ted to a vertical end load F Figure 1(b) shows the pseudo-rigid-body equiva lent of the straight cantilever beam. The distance from the fixe d end to the beam end in the x direction is a the distance from the fixed end to the beam end in the y -direction is b length of the straight beam is l, is the pseudorigid-body angle and is the characteristic radius factor. The angle of inclination of the beam at the beam end is given by 0. (a) Compliant (b) PRBM equivalent Figure 1: A PRBM for a cantilever beam with a vertical end load (Howell 2001)

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8 The co-ordinates of the beam end of the compliant beam are given in terms of the PRB angle, as: )] cos 1 ( 1 [ l a (1.1) sin l b (1.2) Where =0.85 for a vertical end load. The relationship between and 0 is given by: 24 10 (1.3) These relations are accurate to less than 0.5% error for <64.3o. These rigid-body link equations he lp us to calculate the precise motion of the compliant cantilever i.e. for a given pseudo-rigid-body angle, we can calculate the final co-ordinates of the beam end from the fixed end, a in the x -direction and b in the y -direction. We can also calculate the angle of inclination of the beam, 0. There are analogies between planar mechanisms and spherical mechanisms that make it possible to develop a spherical PRBM from the planar PRBM of a cantilever with a vertical end load. A key component of the analogies between planar and spherical mechanisms is that straight lines in planar mechanisms become great circles or circular arcs in spherical mechanisms (Chiang 1992). Also, angles between lines become angles be tween planes (containing great

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9 circles). For example, a planar me chanism may have an input in the y direction and an output in the x -direction as shown in Figure 2. Figure 2: Planar Mechanism with slid ers moving on perpendicular straight lines The analogous spherical mech anism will travel on two perpendicular circular arcs Ydirection ( equivalent of y direction) and an output in the X -direction (equivalent of x-direction) as shown in Figure 2 as shown in Figure 3.

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10 Figure 3: Spherical mechanism with slid ers moving on perp endicular circular arcs Note that spherical mechanisms whose size is very small compared to the radius of the sp here closely approximate planar mechanisms. In fact, spherical kine matics is identical to the planar kinematics in the limiting case wh en the radius of the sphere is infinite. We are also motivated by the ideas that relate planes and spheres such as the stereographic projections used by cartographers to represent a spherical earth on a flat map or the mathematical identification between th e complex plane and the Riemann sphere (Frankel 1997). Let us divide the sphere S just like the earth into latitudes, longitudes and equator. All longitudes and the equator are great circles (WordNet 2001). Great ci rcles are circles that have the

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11 same radius as the sphere and define a plane which cuts the sphere into two equal halves (Merriam-W ebster Dictionary 2006). For example great circles on the surface of the ea rth have their radius equal to the radius of the earth. A great circle is also the shortest path between any two points on the surface of a sphere. The shortest line between two points on a mathematically defined surface is called a geodesic (Henderson 1998). A geodesic is a stra ight line on a plane and a great circle on sphere. On a sphere all and only great circles are geodesics (on the earth only longitudes an d the equator are great circles (geodesics), latitudes other than th e equator are not great circles and hence latitudes (except the equator) are not geodesics). Thus a great circle on a sphere is analogous to a straight line on a plane. (a) Geodesic on a plane Figure 4: Geodesics

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12 (b) Geodesic on a sphere Figure 4: (Continued) Figure 4(a) shows the shortest path p between A and B on a plane. Figure 4(b) shows the shortest path p between A and B on a sphere. Moreover, on a sphere be cause “straight” lines are great circles (curved), there are no parallel lines. ‘Parallelism’ does not exist, that is, all great circles intersect on a sphere. Parallel transport on a sphere is an analogous concept to pa rallel lines on a plane. Lines that intersect a geodesic (great circles) with the same angle are parallel transports (Henderson 1998).

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13 Figure 5: Parallel tr ansport (Henderson 1998) We use parallel transport to understand how forces and displacements should be applied to a spherical mechanism in a way that is analogous to a vertical disp lacement in a planar mechanism. In spherical mechanisms, force and ve locity vectors in a particular tangent plane should continue to be tangent to the sphere and follow the motion of the mechanism. Henc e the force and velocity vectors need to change direction as th e mechanism moves. On a sphere, different tangent planes have differe nt normal vectors. For the force and velocity vectors to be in the tangent plane, any normal component of the vector must be removed. Parallel transport of a vector along a longitude (great circle) can be found by copying the original vector and removing the normal comp onent (Henderson 1998).

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14 A vector m is parallel transported along a longitude to obtain a vector n as shown in Figure 6. Figure 6: Parallel transport along the same longitude All longitudes make the same an gle with the equator (geodesic) (Henderson 1998). Thus, all longitudes are parallel transports of each other. At any point in the northern hemisphere, all vectors pointing to the North Pole will lie on longitudes thus any vector in the northern hemisphere and pointing towards the North Pole is a parallel transport of any other such vector. Thus, a no rthward-pointing force-vector on the equator of a sphere can be paralle l transported to a vector pointing north at any other point on the sphere. A vector pointing north on a sphere is analogous to a ve rtical force in a plane.

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15 1.3 Roadmap This chapter has presented back ground on PRBMs and spherical kinematics, Later chapters describe, how the spherical PRBM is modeled, analyzed and validated. Chapter 2 describes the analogy between planar PRBM and a sphe rical PRBM. It also gives the nomenclature and topology for the spherical PRBM. Chapter 3 describes the finite element mode l and how the displacements were applied to the model. Chapter 4 de scribes how the data was used to obtain the values for the PRBM parameters given in the second chapter. Chapter 5 describes the resu lts obtained for different aspect ratios, b/h and arc lengths, Chapter 6 is the conclusion based on the results.

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16 2. Methodology and Model Development 2.1 Correspondence between spherical and planar PRBMs Mechanisms whose joint axes ar e parallel to each other are known as planar mechanisms (C hiang 1992). In planar compliant mechanisms, this characteristic is usually achieved by designing straight cantilevers (flexures) that, at each point along their length, are most flexible about parallel lines and considerably more rigid in other directions. Mechanisms whose jo int axes intersect at a point are spherical mechanisms (Chiang 1992). In spherical compliant mechanisms, this characteristic ca n be achieved by designing curved cantilevers (flexures) that, at ea ch point along the arc, are most flexible about lines that point to the centre of the sphere. In both kinds of mechanisms it is necessary that the length (arc-length) of flexure be much greater than the width of the beam (flexure), and the width of the beam to be larger than its thickness. It is hypothesized that a flexure which is a long, thin circular arc will move in a manner consistent with spherical kinematics when loaded appropriately. The proces s of obtaining the PRBM for a

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17 spherical compliant mechanism is similar to planar compliant mechanism. Figure 7: Relationship between existing planar PRBM and the spherical PRBM developed in this work The spherical compliant mech anism and its rigid body counterpart are derived from th e planar mechanism by making straight lines curved. There is a correspondence principle between spherical PRBMs and planar PRBMs. The correspondence principle is that when small angle assumption is used for spherical arcs. i.e. the arc length is much smaller than the radius of the sphere, the spherical PRBM becomes identical to planar PRBM. To emphasize the relationship between lines and arcs, the lengths in planar model are denoted with Roman letters, and the equivalent arcs in the spherical model are denoted with the Greek le tter equivalents. For example the

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18 arc length, that appears in some formulas for spherical mechanisms, can be related to the pl anar length, b. Thus, using small angle approximation. b sin 1 cos Where b is the planar equivalent of the arc Similarly a and lare the planar equivalent of arcs and respectively. Additionally, similar terminology is used in planar and spherical PRBMs, for angles between lines (arcs) such as 0, and for ratios such as and C. These variables do not change in the small angle case. In the planar case, the deflected angle of beam end, 0, is about an axis normal to the plane. Simi larly, in the spherical case, the deflection of the beam end, 0, is about an axis normal to the tangent plane to the sphere at the beam end. 2.2 Kinematics of compliant circular arc The kinematics of the compliant circular cantilever, PQ is described by using a series of co-o rdinate frames, as shown in Figure 8. The fixed end of the curved cantilever beam is denoted as P and free end of the beam as Q Let S be a sphere whose center is defined by O frame and the frames A B C and D are always on the surface of

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19 the sphere. The position and orient ation of the co-ordinate frames are related as follows: The O frame is a fixed frame that locates the center of the sphere. The A frame is a frame that locates the beam end Q in undeflected co-ordinates with neutral axis of beam at Q is parallel to the a3 direction and the a1 direction is outward radial vector through the beam end. The B frame is a frame that locates the deflected position of the beam end Q in the x-z plane (analogous to the translation in the x direction in the planar model). The C frame is a moving frame that describes movement of beam end Q in the b2-b1 plane rotating about point O (analogous to the translation in the y-direction in the planar model). The D frame is a moving frame at the same position as the C frame and tracks the deflection of the beam end about the radial axis through the beam end (analogous to the deflection about the z -axis in the planar model).

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20 Figure 8: Reference frames describing the motion of the end of a compliant circular cantilever The frames are described by the matrices A B C and D where the columns of the matrix are the basis vectors. The transformations relating the frames are given by: 1 0 0 0 1 0 0 0 1 }] { }, { }, [{3 2 1a a a A A a R A b b b B ) ˆ ( ) cos( 0 ) sin( 0 1 0 ) sin( 0 ) cos( }] { }, { }, [{2 3 2 1

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21B b R B c c c C ) ˆ ( 1 0 0 0 ) cos( ) sin( 0 ) sin( ) cos( }] { }, { }, [{3 3 2 1 C c R C d d d D ) ˆ ( ) cos( ) sin( 0 ) sin( ) cos( 0 0 0 1 }] { }, { }, [{0 1 0 0 0 0 3 2 1 The transformations relating the frames are given by: A a R b R c R D ) ˆ ( ) ˆ ( ) ˆ (2 3 0 1 Thus the motion of the cantilever beam is described by the parameter = -, and 0 which are analogous to planar parameters l-a b and 0, respectively which are shown in Figure 1. 2.3 Spherical kinematics of the pseudo-rigid-body model Now by analogy to the planar PRBM, in the spherical PRBM, is defined as the pseudo-rigid-body angle of the beam end about the characteristic-pivot (pseudo-pivot) and is defined as the ratio of the arc length from the beam end to the pseudo-pivot to the entire arc length of the beam. The value of is chosen so that the motion of the beam end closely approximates the motion of the compliant beam. The details of selecting the value of are explained in chapter 4. Thus the proposed topology for the pseudo-rigid-body model is shown in Figure 9.

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22 Figure 9: The pseudo-rigid-body mo del of the compliant curved beam The relationships for and in terms of and are obtained using Napier rules for right spherica l triangle (Spiegel, 1968). The right spherical triangle in Figure 9 has sides , and (See Appendix A) where ) 1 ( Thus we find as a function of and ) cos (tan tan ) 90 tan( tan ) 90 sin(1 (2.1)

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23 And is obtained as ) cos (tan tan ) 1 ( ) 1 (1 (2.2) Also is obtained as a function of and ) sin (sin sin sin sin sin1 (2.3) 2.4 Spherical loading condition an alogous to planar vertical end load Based on the discussion in chapter 2, the spherical equivalent of a vertical end load is northwardpointing end-load. An important distinction between planar and spherical loading conditions is that the planar load direction is constant; the spherical load direction must change. A vertical end load in the planar case always points upward, on a sphere there is no such one direction to which the load vector points. The direction of the forc e vector should change as the mechanism moves along the curvature of the sphere. In practice the change requires that any component of force in the direction normal to the sphere must be removed, perhaps by addition of load bearing members in the mechanism. Thus, at any other point on the sphere the vector initiating from that po int and pointing towards the North Pole imitates a vertical end load in planar case.

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24 3. Finite Element Analysis (FEA ) To deduce the accurate motion of the beam going through spherical motion the beam is modele d in a FEA software package. The parametric angle co-efficient, C, the characteristic radius factor, and the parameterization limit, max, are obtained from the results of the FEA model. A major challenge in building the model in FEA package is to apply loads on the beam such that there is no reaction load at the fixed end, P (see Figure 10) and the free end, Q of curved cantilever beam moves in a manne r consistent with spherical kinematics. For this study we fo cus on the motion of the beam (kinematics), the reaction loads will be studied in later work. Development of the model is a paradox because the load direction depends on the displace ment of the beam end, and the displacement of the beam end depend s on the load direction. Thus, to ensure that there is no reac tion load at the fixed end, P we need to know the path (dotted line shown in Figure 10) followed by the beam end. The path followed is an arc on the sphere from the A frame (undeflected position Q ) to the C or D frame (final position Q’’ ).

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25 Figure 10: Path followed by beam the dotted line from Q to Q’’ When the beam PQ is taken as fixed at P the A -frame of reference is fixed. The motion of th e beam can also be described in the B -frame of reference such that the end Q of the beam is allowed to move in the b1-b2 plane. As a consequence of this the end P of the beam now moves in the b1-b3 plane, that is, the beam undergoes spherical motion such that the ends P and Q move on orthogonal great

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26 circles. To illustrate clearly, the co mparison with the planar case is shown. (a) Planar Fixed reference frame (b) Planar Moving reference frame (c) Spherical Fixed reference frame (d) Spherical Moving reference frame Figure 11: Reference frames used to model the spherical mechanism and its planar equivalent We can see from Figure 11(a) and 11(c) that if an input is given at the free end Q, the output obtained when the beam is fixed at P is a displacement at Q On the other hand, in Figure 11(b) and 11(d) an

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27 input is given at Q and the output is obtained at P. As we see from Figure 11, the difference between the fixed frame of reference and moving frame of reference is the loca tion of the output displacements. In the planar case, when an input displacement of b is given the output obtained is o=l-a in both the fixed frame of reference, shown in Figure 11(a), and the moving frame of reference Figure 11(b). In the spherical case, when an input displacement of is given, the output obtained is = in both the fixed frame of reference, shown in Figure 11(c), and the moving frame of reference, shown in Figure 11(d). Thus, the mechanisms are equi valent to each other and only the frame of reference has changed. It proves convenient to analyze the behavior of the flexible curved beam in a FEA model built to mimic the moving frame. In this frame of reference, we apply displacement loads at Q and measure the output displacement at P The fixed frame of reference is the A frame in Figure 10 In order to get the spherical frame B When the B -frame is observed in a movi ng frame of reference it coincides with the A -frame for all northward-pointing input displacements. The initial position is such that both the ends of the beam are in the b1-b3 plane. An input of displacement angle, is applied to the beam end Q The motion of Q is a circular arc in b1-b2 plane. The output obtained is the displacement angle (about the y

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28 axis of the O frame) observed at the other end P of the beam. The motion of this beam end, P, is a circular arc that lies in the b1-b3 plane. The mechanism shown in 11(d) is mo deled in FEA software package. The ANSYS version 10.0, (ANSYS, 2006) FEA package was used. A major aspect of modeling in ANSYS is that it do es not take inputs or outputs with respect to units. Hence the model itself has to be built in a single system of consistent units. Since this is a ‘kinematic’ model, the factors expected to affect the results would be dimensions of the curved beam and Modulus of Elasti city. Here the model dimensions were defined in millimetres (mm) an d the modulus of elasticity in Newton per square millimetre (N/mm2). In this model we take the length of the rigid beam OP =1000mm, the radius of the arc PQ =1000mm. The Q end of the beam is always in the X-Y plane and its initial position for all arc-lengths of PQ, is Q(1000,0,0). The initial position of the end P varies for different arc-lengths and is given by P(R*co s(arclength),0,R*sin(arclength)). Where R is the radius of th e arc (sphere) =1000mm and arclength is the angle created by th e arc to the centre in radians. The mechanism is modeled such that the circular segment PQ is highly compliant and the straight segment OP is highly rigid. This is done by maintaining the modulus of el asticity of the compliant circular segment at 300N/mm2 and that of the rigid straight segment at

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29 300,000 N/mm2. Various aspect ratios of the beam are obtained by varying the cross-section of the beam th at is if an aspect ratio of 0.1 is desired then the thickness (or height h ) of the beam is 1/10 th of the width b When aspect ratio of the beam is 1 the beam has a square cross-section of sides 50mm, for successive values of aspect ratio the sides vary accordingly to obtain a rectangular cross-section of width b and height h given by h =aspect ratio b as shown in Figure 12. Figure 12: Cross-section of be am for various aspect ratios This model is then meshed to define elements and nodes. Displacement loads are applied acco rding to the boundary conditions described below. To apply the boundary conditions for the above model we denote the displacements in x y and z directions by UX, UY and UZ and

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30 rotations about x y and z by ROTX, ROTY and ROTZ. Points O and P lie on the rigid straight segment and hence they are made to stay in the x-z plane and allowed to rotate about y -axis of O frame. The point O fixes the structure in space and hence all other degrees of freedom are constrained. The point Q is the end of the curved segment and hence it is made to lie in the x-y plane. The boundary conditions applied to the finite element model shown in Figure 13 are: Point O UX=0, UY=0, UZ=0, ROTX=0, ROTZ=0. Point P UY=0, Point Q UZ=0, ROTZ= ROTX=0, ROTY=0. Figure 13: Finite element model

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31 Rotational displacement loads, were applied at the Q end of the beam and analysis was cond ucted. For various inputs of we get corresponding outputs of = -. The deflection 0 of the neutral axis of the beam at beam end (that moves in b1-b2 plane) about the radial axis of the beam at the same beam end is also obtained as an output. The deflection of the beam is calc ulated from the rotation matrix generated by a pre-defined triad at the Q end of the beam. Thus these outputs are noted for various inputs and this simulation is repeated for varying: a) Initial arc length b) Cross-section of the curved flexible beam. The results from FEA mode l were used to calculate the parametric angle co-efficient, C, the characteristic radius factor, and 0max. See Appendix C for a visual manual to conduct one simulation. A log file generated from one analysis (simulation) is then obtained from the file menu. This log file is th en edited with new values of the parameters (aspect ratio and arclength) and subsequently run in ANSYS by using an input command Simulations were run for arclengths ranging from angle of 4 degr ees to 112 degrees in increments of 2 degrees and for each arclength as pect ratios varying from 0.1 to 1 with increments of 0.1. The input displacement, is given such that it is equal to the angle created by the re spective arc-length, that is, if an

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32 arc-length of 90 degrees is to be an alyzed then an input displacement of 90 degrees is applied.

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33 4. Parametric Approximation of the Curved Beam’s Deflection Path We follow Howell’s method (Howell 2001) for developing our parametric approximation of the curved beam’s deflection path. An acceptable value for the characteristic radius factor, may be found by first determining the maximum acceptable percentage error in deflection. The value of that would allow the maximum pseudo-rigidbody angle, while still satisfying the maximum error constraint is then determined. The prob lem may be formally stated as follows: Find the value of the characteristic radius factor which maximizes the pseudo-rigid-body angle, where for a spherical mechanism is derived from Napier Rules. For right spherical triangle whose sides are , and and is the angle between and it can be shown that: cot tan )] (1 sin[ (See Appendix A) where to get )] 1 ( sin[ tan tan1 (4.1)

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34 Equation (4.1) is valid for cot tan sin sin tan1 (4.2) Equation (4.2) is applicable for all values of (See AppendixA) and is subject to the parametric constraint (4.3) where error/ e is the relative deflection error, and e for a spherical mechanism is defined as the vector di fference of deflected position of the flexible curved segment and the original un-deflected position. ) 0 ( ) / ( / ) (max max for error error ge e

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35 Figure 14: Deflection of curved segment The deflection of curved segment e as shown in Figure 14 is obtained using finite element analys is software. For various values of the corresponding values of are noted. These are then used to calculate the final position from ro tation and transformation matrices. Finally the original co-ordinates of the beam end are subtracted from the final co-ordinates to obtain the deflection e.

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36 sin cos sin 1 cos cos 0 0 1 0 0 1 ) cos( 0 ) sin( 0 1 0 ) sin( 0 ) cos( 1 0 0 0 cos sin 0 sin cose (4.4) to get (4.5) and the deflection for the PRBM, a, is given by the vector difference of deflected position of the PRBM and the original un-deflected position of the beam end Q sin cos sin 1 cos cosez ey ex e

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37 Figure 15: Deflection of PRBM The vector difference between the estimated deflected position of PRBM and the original un-def lected position of beam end, Q is calculated using the following transformations. From Figure 15 we have r Ra and r R R where R is the rotation of the vector r about the axis m through angle Lai, Rubin and Krempl, 1993

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38) ( sin cos ) )( cos 1 ( ) , ( r m r m r m r m R where, ˆ sin 0 ˆ cos m and 0 0 1 r which reduces to ) cos 1 ( ˆ sin ˆ cos sin ˆ sin cos ) cos 1 ( ˆ cos2 r R R Therefore, ) cos 1 ( ˆ sin ˆ cos sin ˆ sin 1 cos ) cos 1 ( ˆ cos2 a (4.6) error is simply defined as the vect or difference between the final positions of the curved flexible se gment and the pseudo-rigid body model.

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39 The erro r in the deflection is calculated as 2 / 1 2 2 2 az ez ay ey ax ex a eerror (4.7) A parameter relative error, error/ e is defined to help in comparing with the planar flexible segment. e a e eerror (4.8) The value of the angular deflection of the beam’s end, 0, at the point at which the error equals or exceeds an acceptable amount, is the maximum angular deflection of the beam’s end, or the parameterization limit max.

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40 5. Results and Discussion For a given value of aspect ratio, h/b and arc-length, one can find a value of characteristic radius factor, and parametric angle coefficient, C that best approximates the motion (position and orientation of beam at various input displacements) using the techniques described in the previous chapter. For example for h/b =1, and the final displacement of beam from the fixed end, and the rotation 0 are found for a given input displacement of They are plotted against as shown in Figure 16 and Figure 17.

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41 Figure 16: Final position of beam from fixed endv/s input displacement Figure 17: Deflection of beam about neutral axis, 0,v/s input displacement

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42 Then the values of and C that gives the minimum relative error (0.05%) for the largest range of different guess of are found for maximum range of motion. C is the parametric angle co-efficient, defined as the ratio of the maximum range of motion obtained, max, in the pseudo-rigid body model to the ratio of the deflecti on of the beam about the neutral axis, 0. The and C obtained for all the simu lations of various aspect ratios and arc lengths are plotted in Figure 18 and Figure 19 and in Figure 20 the maximum range of motion for the PRBM for the respective values of and C. Figure 18: v/s Arc-lengths showing various colors for aspect ratios

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43 Figure 19: C v/s Arc-lengths showing variou s colors for aspect ratios Figure 20: max v/s Arc-lengths showing various colors for aspect ratios

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44 These are graphs that are plotte d for simulations which were run to obtain outputs at every one degree of the input displacement, that is, if the beam was given a total input displacement of 90 degrees then 90 load-steps of one degree were solved. As a consequence not enough data points were obtained when the total input displacement, was a small value, for example, if the beam was given a total input displacement of just 10 degrees then only 10 loadsteps were solved. Thus, the algorith m to process the outputs obtained from the simulations failed to proce ss the data for arc-lengths ranging from 4 to 14 because the total input displacement, is given such that it is equal to the arc-length. Hence, the number of data points at an arc-length, were limited to the value of arc-length in degrees, that is, only 4 data points were obtained for an arc-length of 4 degrees. Moreover, from the graphs it can be seen that there is a lot of ‘bouncing’ that is there is a ‘noise ’ in the data. This clearly indicates that more data points are required to capture the behavior of the curved beams. Based on the inference of these gr aphs the simulations were re-run such that 200 load-steps are solved irrespective of the value of the input displacement, that is, for an input displacement of 4 degrees the beam was analyzed at an input displacement, of every 4/200 degrees. These simulations were run for aspect ratios h/b =0.1, 0.4

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45 and 0.7. These are then again plotte d as shown in Figure 21, Figure 22 and Figure 23. Figure 21: v/s Arc-lengths, for 200 load-steps of input displacement

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46 Figure 22: C v/s Arc-lengths, for 200 load-s teps of input displacement

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47 Figure 23: max v/s Arc-lengths, for 200 load-s teps of input displacement From Figures 21, 22 and 23 we see th at there is no ‘bouncing’ or ‘noise’ in the data, smoot h curves are obtained. It is observed that this data is suitable to approximate the motion of the beam. An equation is fitted to the curve for the characteristic radius factor and can be used to approximate the motion of a curved beam with the equivalent of vertical end load. A trend-line of second order polynomial for arclengths ranging from 16 to 112 is fit individually for aspect ratios h/b =0.1, 0.4 and 0.7 and their equa tions are shown in Figure 24, Figure 25 and Figure 26.

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48 y = -7E-06x2 + 0.0002x + 0.8480.77 0.78 0.79 0.8 0.81 0.82 0.83 0.84 0.85 0.86 020406080100120 Arclength, Characteristic radius factor, Figure 24: Trend-line of for aspect ratio 0.1 y = -7E-06x2 + 0.0001x + 0.85070.77 0.78 0.79 0.8 0.81 0.82 0.83 0.84 0.85 0.86 020406080100120 Arclength, Characteristic radius factor, Figure 25: Trend-line of for aspect ratio 0.4

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49 y = -6E-06x2 1E-04x + 0.85760.77 0.78 0.79 0.8 0.81 0.82 0.83 0.84 0.85 0.86 0.87 020406080100120 Arclength, Characteristic radius factor, Figure 26: Trend-line of for aspect ratio 0.7 Thus, at a given aspect ratio h/b and arc-length of curved beam we can substitute the values in the respective equation to find the corresponding characteristic radius factor for the spherical PRBM that best approximates the moti on of the curved flexible beam.

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50 6. Conclusion The first Pseudo-Rigid-Body Model (PRBM) for spherical mechanisms has been developed. The kinematics of a compliant curved beam and its rigid body equivalent were described. The procedure for analyzing the curved compliant beams in a FEM program was developed. Pseudo-rigid body parameters were calculated from FEA results. These parameters are the characteristic radius factor, the parametric angle co-efficient C and the parameterization limit max. These values approach the values found in the planar case for small arc lengths,

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51 References 1) Ananthasuresh, G.K., 1994, “A New Design Paradigm for MicroElectro-Mechanical Systems and In vestigations on the Compliant Mechanism Synthesis,” Ph.D. di ssertation, University of Michigan, Ann Arbor, MI. 2) Ananthasuresh, G. K., and Kota, S., 1995, “Designing of Compliant Mechanisms,” Mechanical Engineering Vol. 117, No.11, pp. 93-96. 3) Ananthasuresh, G.K., Kota, S ., and Gianchandani, Y., 1994, “A Methodical Approach to the Design of Compliant Micromechanisms,” Solid-State Sensor and Actuator Workshop Hilton Head Island, SC, pp. 189-192. 4) ANSYS 10.0, ANSYS University Advanced Release 10.0 5) Baker, M.S., Lyon, S.M., an d Howell L.L., 2000, “A Linear Displacement Bistable Micromechanism,” Proceedings of the 26th Biennial Mechanisms and Robotics Conference 2000 ASME Design Engineering Technical Conferences DETC2000/MECH14119. 6) Boettama.O. and Roth, B., 1979, Theoretical Kinematics Dover, New York. 7) Brocket W. R. and Stokes A., 1991 “On the Synthesis of Compliant Mechanisms” 8) Cannon J.R., Lusk C.P. and Howell, L.L., 2005 “Compliant Rolling-Contact Element Mechanisms”, ASME Mechanisms and Robotics Conference 2005. 9) Chiang, C. H., 1992, Spherical kinematics in contrast to planar kinematics, National Taiwan University, Taipei, Taiwan, Mech Mach Theory v 27 n 3 May 1992 p 243-250.

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52 10) Crane, N.B., Howell L.L., and Weight, B. L., 2000, “Design and Testing of a Compliant Floating-Opposing Arm (FOA) Centrifugal Clutch,” Proceedings of 8th International Power Transmission and Gearing Conference 2000 ASME Design Engineering Technical Conferences DETC2000/PTG-14451. 11) Dado M. H, 2001 “Variable Parametric Pseudo-Rigid Body Model for Large Deflection Be ams with End Loads” International Journal of Non-linear Mechanics 2001. 12) Derderian, J.M., 1996, “The Ps eudo-Rigid Body Model Concept and its Application to Micro Comp liant Mechanisms,” M.S. thesis, Brigham Young University, Provo, UT. 13) Erdman, A.G., and Sandor, G.N., 1997, Mechanism Design: Analysis and Synthesis Vol. 1, 3rd Ed., Prentice Hall, Upper Saddle River, NJ. 14) Frankel, T., 1997 The Geometry of Physics, Cambridge University Press, pp21-22. 15) Frisch-Fay, R., 1962, Flexible Bars Butterworth, Washington DC. 16) Guerinot A. E., Magleby S. P., and Howell, L. L., 2004 “Preliminary Design Concepts for Compliant Mechanism Prosthetic Knee Joints”, ASME Mechanisms and Robotics Conference 2004. 17) Her, I., and Midha, A., 1987, “A Compliance Number Concept for Compliant Mechanisms, and Type Synthesis.” Journal of Mechanisms Transmissions, and Automa tion in Design, Trans. ASME, Vol. 109, No. 3, pp. 348-355. 18) Her, I., 1986, “Methodology fo r Compliant Mechanisms Design,” Ph.D dissertation, Purdue University, West Lafayette, IN. 19) Henderson, D. W., 1998 Differential Geometry Prentice Hall Upper Saddle River, NJ. 20) Hill, T.C., and Midha, A., 1990, “A graphical user-driven NewtonRaphson Technique for use in the Analysis and Design of Compliant Mechanisms.” Journal of Mechanical Design, Trans ASME, Vol. 112, No. 1, pp. 123-130.

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53 21) Howell L. L., 2001, Compliant Mechanisms John Wiley and Sons, Inc, NY. 22) Howell, L. L., 1991, “The Design and Analysis of Large Deflection Members in Compliant Mechanisms,” M.S. thesis, Purdue University, West Lafayette, IN. 23) Howell, L. L., and Midha, A ., 1993, “Compliant Mechanisms,” Section 9.10 in Modern Kinematics: Developments in the Last Forty Years, (A. G. Erdman, ed .), Wiley, New York, pp. 422-428. 24) Howell, L. L., and Midha, A. 1994, “The development of Force Deflection Relationships,” Machine elements and Machine Dynamics: Proceedings of the 1994 ASME Mechanisms conference, DE-Vol. 71, pp. 501-508. 25) Howell, L. L., and Midha, A ., 1995, “Parametric Deflection Approximations for End-Loaded Large Deflection Beams in Compliant Mechanisms,” Journal of Mechanical Design ASME, Vol. 117, No.1, pp. 156-165. 26) Howell, L. L., and Midha, A ., 1995, “Determination of the Degrees of Freedom of Compliant Mechanisms using the PseudoRigid Body Model Concept,” Proceedings of the 9th World Congress on the Theory of Machines and Mechanisms Milano, Italy, Vol. 22, pp. 1537-1541. 27) Howell, L.L., Rao, S.S., and Midha, A., 1994, “The Reliability Based Optimal Design of a Bistable Compliant Mechanism,” Journal of Mechanical Design Trans. ASME, Vol. 116, No. 4, pp. 1115-1121. 28) Howell, L.L., and Midha, A. 1995, “Parametric Deflection Approximations for End-Loaded Large-Deflection Beams in Compliant Mechanisms,” Journal of Mechanical Design Trans. ASME, Vol. 71, pp. 156-165. 29) N.B. Hubbard, L.L. Howell, 2005 “Design and Characterization of a Duel-stage, Thermally Actuated Nanopositioner”, Journal of Micromechanics and Microengineering, 2005.

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54 30) Jensen, B.D., Howell L.L., Gunyan, D.B., and Salmon, L.G., 1997, “The Design and Analysis of Compliant MEMS Using the Pseudo-Rigid-Body Model,” Micromechanical Systems (MEMS), at the 1997 ASME International Mech anical Engineering Congress and Exposition DSC-Vol. 62, pp. 119-126. 31) Jensen, B.D., Howell L.L., an d Salmon, L.G., 1999, “Design of Two-Link, In-plane, Bi-stable Compliant Micro-mechanisms,” Journal of Mechanical Design Trans. ASME, Vol. 121, No. 3, pp. 93-96. 32) Lai, M.W., Rubin, D, Krempl, E, 1993 Introduction to Continuum Mechanics 3rd ed Pergamon, Pr. 33) Merriam-Webster Dictionary, 2006, Merriam-Webster Online. “ http://www.m-w.com/cgibin/dictionary?book=Dictionary&va=great%20circle”. 34) Midha, A., Her, I., and Salamo n, B.A., 1992, “A Methodology for Compliant Mechanisms Design, Pa rt I: Introduction and LargeDeflection Analysis,” in Advances in Design Automation, D.A Hoeltzel, ed., 18th ASME Design Automation Conference DE-Vol. 44-2, pp. 29-38. 35) Midha, A., and Howell, L.L., 2000, “Limit Positions of Compliant Mechanism using the Pseudo-rigid body Model”, Mechanism and Machine Theory Vol. 35, No. 1, pp. 99-115. 36) McCarthy, J., 2000 Geometric Design of Linkages Springer, NY. 37) Millar A. J., Howell L L an d Leonard J N, 1996 “Design and Evaluation of Compliant Constant Force Mechanisms”. 38) Roach, G.M., and Howell, L. L., 2000, “Compliant Overrunning clutch with centrifugal Throw out,” U.S. patent 6,148,979, Nov 21. 39) Salamon, B.A., 1989, “Mecha nical Advantage Aspects in Compliant Mechanisms Design,” in Advances in Design Automation, D.A Hoeltzel, ed., 18th ASME Design Automation Conference, DE-Vol44-2, pp. 47-51

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55 40) Saggere L. and Kota S.,1997 “Syn thesis of Distributed Compliant Mechanisms for Adaptive Structure application: An ElastoKinematic Appproach” Proceedings of the ASME Design Engineering Technical Conferences ,1997. 41) Salmon, L.G., Gunyan, D.B., De rderian, J.M., Opdahl, P.G., and Howell L.L., 1996, “Use of the Pseudo-Rigid Body Model to Simplify the Description of Co mpliant Micromechanisms,” 1996 IEEE Solid-State and Actuator Workshop Hilton Head Island, SC, pp. 136-139. 42) Saxena, A. and Kramer, S.N., 1998, “A Simple and Accurate Method for Determining Larg e Deflections in Compliant Mechanisms Subjected to End Forces and Moments,” Journal of Mechanical Design Trans. ASME, Vol. 120, No.3, pp. 392-400, erratum, Vol. 121, No. 2, p.194. 43) Sevak, N.M., and McLarman, C.W., 1974, “Optimal Synthesis of Flexible Link Mechanisms with large Static Deflections,” ASME Paper 74-DET-83. 44) Spiegel M. R., 1968 Schaums Outlines of Theory and Problems Of Mathematical Handbook Of Formulas And Tables. 45) WordNet “longitude”. WordNet 1.7.1 Princeton University, 2001. Answers.com, 14 Nov. 2006. http://www.answers.com/topic/longitude 46) Wolfram, Mathworld, 2006, http://mathworld.wolfram.c om/SphericalTriangle.htm

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56 Appendices

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57 Appendix A Spherical Tria ngles and Napier Rules Spherical Triangles Figure 27: Spherical triangles A spherical triangle is a figure fo rmed on the surface of a sphere by three great circular arcs intersecti ng pair-wise in three vertices. The spherical triangle is the spherical an alog of the planar triangle, and is sometimes called an Euler triangle (Wolfram, 2006). Let a spherical triangle have angles A, B, and C (m easured in radians at the vertices along the surface of the sphere) and let the sphere on which the spherical triangle sits have radius R (Wolfram, 2006) Napier Rules Napier’s rules are used to derive the parameters required to analyze the bending of curved beam.

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58 Appendix A (Continued) The derivation of parameters ca n be easily obtained from two simple rules discovered by John Napier (1550-1617), the inventor of logarithms. (http://www.angelfire.com/nt/navtrig/B2.html). As the right angle does not enter into th e formulas, only five parts are considered. These are a, b, and th e complements of A, B, and C (or 90-A, 90-B, 90-c) which can be written A', B', and c'. If these five parts ar e arranged in the order in which they occur in the triangle, any part may be selected and called the middle part; then the two parts next to it are called adjacent parts, and the other two are called opposite parts. Figure 28: Five parts arranged in order of occurrence Napier’s rules are as follows: 1. The sine of the middle part equals the product of the tang ents of the adjacent parts.

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59 Appendix A (Continued) 2. The sine of the middle pa rt equals the product of the cosines of the opposite parts. The right spherical triangle for the PRBM has the sides, and The right angle lies between the ‘sides’ and is the pseudo-rigid-body angle. ‘ ’ is the angle opposite to as shown in Figure 24 Figure 29: Spherical right triangle

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60 Appendix A (Continued) Figure 30: Five parts for PR BM right spherical triangle. Using Napier Rules the following equations can be obtained. ) 90 tan( tan ) 90 sin( Where ) ( ) ( and To get )] 1 ( sin[ tan tan1

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61 Appendix A (Continued) At =90o this equation fails to give a value of pseudo-rigid body angle, to overcome this, is also expressed in an alternate form. From Napier Rules we get sin sin sin and cot tan cos To get cot tan sin sin tan1

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62 Appendix B Manual for FEA The following figures (31-54) show a step by step process to run a single simulation for a single load step. Figure 31: Activating Graphical User Interface (GUI)

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63 Appendix B (Continued) Figure 32: Limiting the GUI opti ons to structural preferences

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64 Appendix B (Continued) Figure 33: Adding or defining new element types

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65 Appendix B (Continued) Figure 34: Beam elements

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66 Appendix B (Continued) Figure 35: Defining real cons tants for respective elements

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67 Appendix B (Continued) Figure 36: Inputting area and moment of inertia values to elements

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68 Appendix B (Continued) Figure 37: Defining material properties

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69 Appendix B (Continued) Figure 38: Creating key-points on work-plane through GUI

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70 Appendix B (Continued) Figure 39: Creating key-points using command line

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71 Appendix B (Continued) Figure 40: Pan, zoom, rotate

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72 Appendix B (Continued) Figure 41: Defining of orth ogonal triad at beam end

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73 Appendix B (Continued) Figure 42: Creating lines

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74 Appendix B (Continued) Figure 43: Creating arcs

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75 Appendix B (Continued) Figure 44: Meshing

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76 Appendix B (Continued) Figure 45: Mesh attributes for line

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77 Appendix B (Continued) Figure 46: Allocating specific material to mesh (elements)

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78 Appendix B (Continued) Figure 47: Selecting analysis type

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79 Appendix B (Continued) Figure 48: Large displace ment analysis selected

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80 Appendix B (Continued) Figure 49: Equation chosen solvers

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81 Appendix B (Continued) Figure 50: Applying loads to the beam

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82 Appendix B (Continued) Figure 51: Solve

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83 Appendix B (Continued) Figure 52: During solve

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84 Appendix B (Continued) Figure 53: Output

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85 Appendix B (Continued) Figure 54: To get the log-file A log file is obtained and is modified to solve for 200 load steps at a given aspect ratio and arc-length as follows: !************************************ /CONFIG,NRES,10000 /CWD,'C:\Documents and Settings\sjagirda\Directory200steps\arc90_asp0.1' /NOPR /PMETH,OFF,0

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86 Appendix B (Continued) KEYW,PR_SET,1 KEYW,PR_STRUC,1 /GO !************************************ /PREP7 R=1000 PI=acos(-1.) !************************************ A1=2500.0 Iy1=520833.333 Iz1=520833.333 E1=300000 !************************************ A2= 250.0000037252903 Iy2= 520.8333566163981 Iz2= 52083.33410943548 E2= 300 !************************************ ET,1,BEAM4 !* ET,2,BEAM4

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87 Appendix B (Continued) !* R,1,A1,Iy1,Iz1, , RMORE, , , , !* R,2,A2,Iy2,Iz2, , RMORE, , , , !* !* MPTEMP,,,,,,,, MPTEMP,1,0 MPDATA,EX,1,,E1 MPDATA,PRXY,1,,0.35 MPTEMP,,,,,,,, MPTEMP,1,0 MPDATA,EX,2,,E2 MPDATA,PRXY,2,,0.35 !************************************ K,1,0,0,0, !************************************ arclength=90 xcoor=R*cos(arclength*PI/180)

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88 Appendix B (Continued) zcoor=R*sin(arclength*PI/180) K,2,xcoor,0,zcoor, !************************************ K,3,1000,0,0, K,4,1050,0,0, K,5,1000,50,0, K,6,1000,0,-50, /USER, 1 /FOC, 1, 538.256940599 -110.688686131 475.000000000 /REPLO /VIEW, 1, -0.246365419055 0.245754775350 0.937501291032 /ANG, 1, -1.91248212175 /REPLO /VIEW, 1, -0.378438950955 0.367160066605 0.849692559630 /ANG, 1, -4.76219842328 /REPLO K,7,950,0,0, /FOC, 1, 427.888273403 -68.9410891494 407.804103704 /REPLO

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89 Appendix B (Continued) /FOC, 1, 456.194825599 -81.6519820718 425.903867008 /REPLO /VIEW, 1, -0.455467710156 0.439135756168 0.774408776203 /ANG, 1, -7.33131293670 /REPLO LSTR, 1, 2 LSTR, 3, 4 LSTR, 3, 5 LSTR, 3, 6 LSTR, 3, 7 !* LARC,2,3,1,1000, FLST,5,5,4,ORDE,2 FITEM,5,1 FITEM,5,-5 CM,_Y,LINE LSEL, , ,P51X CM,_Y1,LINE CMSEL,S,_Y !*

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90 Appendix B (Continued) !* CMSEL,S,_Y1 LATT,1,1,1, , CMSEL,S,_Y CMDELE,_Y CMDELE,_Y1 !* FLST,5,1,4,ORDE,1 FITEM,5,1 CM,_Y,LINE LSEL, , ,P51X CM,_Y1,LINE CMSEL,,_Y !* LESIZE,_Y1, ,10, , ,1 !* FLST,5,4,4,ORDE,2 FITEM,5,2 FITEM,5,-5 CM,_Y,LINE LSEL, , ,P51X

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91 Appendix B (Continued) CM,_Y1,LINE CMSEL,,_Y !* LESIZE,_Y1, ,1, , ,1 !* FLST,2,5,4,ORDE,2 FITEM,2,1 FITEM,2,-5 LMESH,P51X GPLOT CM,_Y,LINE LSEL, , 6 CM,_Y1,LINE CMSEL,S,_Y !* !* CMSEL,S,_Y1 LATT,2,2,2, , CMSEL,S,_Y CMDELE,_Y CMDELE,_Y1

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92 Appendix B (Continued) !* FLST,5,1,4,ORDE,1 FITEM,5,6 CM,_Y,LINE LSEL, , ,P51X CM,_Y1,LINE CMSEL,,_Y !* LESIZE,_Y1, ,100, , ,1 !* LMESH, 6 FINISH /SOL ANTYPE,0 NLGEOM,1 NSUBST,10,0,0 OUTRES,ERASE OUTRES,NSOL,-10 RESCONTRL,DEFINE,ALL,-10,1 FLST,2,1,1,ORDE,1 FITEM,2,1

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93 Appendix B (Continued) !* !************************************ /GO D,1, ,0, , ,UX,UY,UZ,ROTX,ROTZ, FLST,2,1,1,ORDE,1 FITEM,2,2 !* /GO D,2, ,0, , ,UY, , , FLST,2,1,1,ORDE,1 FITEM,2,12 !* !************************************ /GO !************************************ loadsteps=200 *DO,step,1,loadsteps,1 theta=step*arclength/200 /GO DDELE,12,ALL !************************************

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94 Appendix B (Continued) D,12, ,0, , ,UZ, , , FLST,2,1,1,ORDE,1 FITEM,2,12 dispx=-(R-(R*cos(theta*PI/180))) dispy=R*sin(theta*PI/180) D,12, ,dispx, , ,UX, , , FLST,2,1,1,ORDE,1 FITEM,2,12 !* /GO D,12, ,dispy, , ,UY, , , FLST,2,1,1,ORDE,1 FITEM,2,12 !* /GO D,12, ,theta*PI/180, , ,ROTZ, , , LSWRITE,step *ENDDO LSSOLVE,1,loadsteps

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95 Appendix B (Continued) /STATUS,SOLU FINISH !************************************ SAVE,'arc90_asp0.1','db','C:\DOCUME ~1\SJAGIRDA\Directory200steps \arc90_asp0.1' !************************************ *do,i,1,200,1, /POST1 /OUTPUT,arc90_asp0.1,txt,,APPEND SET,,,,,i,,, FLST,5,6,1,ORDE,4 FITEM,5,1 FITEM,5,-2 FITEM,5,12 FITEM,5,-15 NSEL,S, ,P51X PRNSOL,DOF, /OUT *ENDDO *do,i,1,200,1, /POST1

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96 Appendix B (Continued) /OUTPUT,arc90_asp0.1,m,,APPEND SET,,,,,i,,, FLST,5,6,1,ORDE,4 FITEM,5,1 FITEM,5,-2 FITEM,5,12 FITEM,5,-15 NSEL,S, ,P51X PRNSOL,DOF, /OUT *ENDDO *do,i,1,200,1, /POST1 /OUTPUT,arc90_asp0.1BETA,txt,,APPEND SET,,,,,i,,, FLST,5,6,1,ORDE,4 NSEL,S, ,12 PRNSOL,ROT,Z /OUT *ENDDO *do,i,1,200,1,

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97 Appendix B (Continued) /POST1 /OUTPUT,arc90_asp0.1DISPX,txt,,APPEND SET,,,,,i,,, FLST,5,4,1,ORDE,2 FITEM,5,12 FITEM,5,-15 NSEL,S, ,P51X PRNSOL,U,X /OUT *ENDDO *do,i,1,200,1, /POST1 /OUTPUT,arc90_asp0.1DISPY,txt,,APPEND SET,,,,,i,,, FLST,5,4,1,ORDE,2 FITEM,5,12 FITEM,5,-15 NSEL,S, ,P51X PRNSOL,U,Y /OUT *ENDDO

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98 Appendix B (Continued) *do,i,1,200,1, /POST1 /OUTPUT,arc90_asp0.1DISPZ,txt,,APPEND SET,,,,,i,,, FLST,5,4,1,ORDE,2 FITEM,5,12 FITEM,5,-15 NSEL,S, ,P51X PRNSOL,U,Z /OUT *ENDDO *do,i,1,200,1, /POST1 /OUTPUT,arc90_asp0.1PHI,txt,,APPEND SET,,,,,i,,, NSEL,S, ,2 PRNSOL,ROT,Y /OUT *ENDDO FINISH /EOF

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99 Appendix B (Continued) The respective parameters affected by change in aspect ratio like real constants are then changed in this lo g file and run separately to obtain respective outputs. The outputs ar e also limited to the Nodes of interest.

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100 Appendix C Algorithms to Find Algorithm to find for load-steps at every one degree MATLAB Program is as follows. clear all start=16; finish=112; for arclength=start:2:finish counter=(arclength+2-start)/2; for aspect=0.1:0.1:1 countas=round(10*aspect); %Input str1 = []; if round(aspect)==aspect, str1='.0'; end string = ['\arc',num2str(arclength),'_asp', num2str(aspect),str1,'ex.txt'];

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101 Appendix C (Continued) fid = fopen(['C:\Documents and Settings\sjagirda\output\arc',n um2str(arclength),string]); A = fread(fid); fclose(fid); G = native2unicode(A)'; s_i = findstr('ROTZ', G); s_f = findstr('MAXIMUM', G); cr = native2unicode(10); space = native2unicode(9); for j = 1:length(s_i) M = strtrim(G(s_i(j)+4:s_f(j)-1)); M = strrep(M, cr, space); M = str2num(M); beta(j) =M(3,7); PHI(j) =M(2,6); B = [ cos(be ta(j)) sin(beta(j)) 0 ; -sin(beta(j)) cos(beta(j)) 0 ; 0 0 1 ]; new cs = [ 50 0 0 ; 0 50 0 ; 0 0 -50 ]; dispatbeta=[ M(3,2) M(3,3) M(3,4) ; M(4,2) M(4,3) M(4,4) ;

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102 Appendix C (Continued) M(5,2) M(5,3) M(5,4) ; M(6,2) M(6,3) M(6,4) ]; orgcoor d = [ 1000 0 0; 1050 0 0; 1000 50 0; 1000 0 -50]; Finalcoord=dispatbeta+orgcoord; node12=[ Finalcoord(1,1:3);Finalcoord(1,1:3) ;Finalcoord(1,1:3);Finalcoord(1,1: 3)]; positi on_vectorbeta=Finalcoord node12; position_vectorbeta(1,:) = []; A = B*newcs*inv(position_vectorbeta); thetaobeta(j)=acos(A(2,2)); end %plot(PHI) beta(j) =M(3,7); PHI(j) =M(2,6); lambda=arclength*pi/180; BG=zeros(arclength,151);

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103 Appendix C (Continued) Beta=0; for countk=1:1:301 for countBETA=1:1:arclength oldbeta=Beta; newbeta=beta(1,countBETA); if ne wbeta==oldbeta countBETA=countBETA-1; break; else Beta=newbeta; gamma=(countk/2000+.7495); gamma_l = gamma*lambda; phi=PHI(1,countBETA); captheta = atan(tan(Beta)./sin((lambda-phi)(lambda-gamma_l))); abs_epsilo n_e = sqrt((cos(Beta).*cos(phi)ones(size(Beta))).^2+(sin(Beta)).^2 .*(cos(phi)).^2+(sin(phi)).^2); epsilon _ex = cos(Beta).*cos(phi)-1; epsilon_ey = sin(Beta); eps ilon_ez = cos(Beta).*sin(phi);

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104 Appendix C (Continued) epsilo n_ax = (cos(gamma_l)).^2.*(1cos(captheta))+cos(captheta)-1; epsilon_ay = sin(captheta).*sin(gamma_l); epsilon_az = sin(gamma_l).*cos(gamma_l).*(1cos(captheta)); error = sqrt((epsilon_ex-epsilon_ax).^2 +(epsilon_ey-epsilon_ay).^2 +(epsilon_ez-epsilon_az).^2); rel _error = error./abs_epsilon_e; captheta 1(countBETA,countk)=captheta; abs _epsilon_e1(countBETA,countk) = abs_epsilon_e; epsilon_ex1( countBETA,countk) = epsilon_ex; epsilon_ey1( countBETA,countk) = epsilon_ey; epsilon_ez1( countBETA,countk) = epsilon_ez; epsilon_ax1( countBETA,countk) = epsilon_ax; epsilon_ay1( countBETA,countk) = epsilon_ay; epsilon_az1( countBETA,countk) = epsilon_az; error1(co untBETA,countk)=error; rel_erro r1(countBETA,countk)=rel_error; if rel_error <= 0.005

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105 Appendix C (Continued) error1(countBETA,countk)=error; rel_e rror1(countBETA,countk)=rel_error; BG(co untBETA,countk)=countBETA*countk; betamax(countk) = Beta; maxf ortheta(countk,countBETA)=Beta; else break end end end end [y,i] = max(betamax); gammastar = (i/2000+.7495); gammastar_l=lambda*gammastar; [p,q]=max(maxfortheta); [r,s]=max(q); Beta=beta(1,s); phi=PHI(1,s); thetaostar=thetaobeta(1,s);

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106 Appendix C (Continued) capthetast ar =atan(tan(Beta)./sin((lambda-phi)-(lambdagammastar_l))); CT HETAstar=capthetastar/thetaostar; GAMMA _MATRIX(counter,countas)=gammastar; CAPTHETA _MATRIX(counter,countas)=capthetastar; CTHETA _MATRIX(counter,countas)=CTHETAstar; end end GAMMA_MATRIX; CTHETA_MATRIX; CAPTHETA_MATRIX; figure(1) plot(GAMMA_MATRIX) figure(2) plot(CTHETA_MATRIX) figure(3) plot(CAPTHETA_MATRIX)

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107 Appendix C (Continued) Algorithm to find for 200 load-steps. clear all start=4; finish=112; for arclength=start:2:finish arclength counter=(arclength+2-start)/2; asp=[0.1 0.4 0.7]; for i=1:3 aspect=asp(i); aspect; countas=round(10*aspect); %Input str1 = []; if round(aspect)==aspect, str1='.0'; end string = ['arc',num2str(arclength),'_ asp',num2str(aspect),str1];

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108 Appendix C (Continued) fi d1 = fopen(['C:\Documents and Settings\sjagirda\Directory200steps \',string,'\',string,'BETA.txt']); ABT = fread(fid1); fclose(fid1); GBT = native2unicode(ABT)'; s_iB = findstr('ROTZ', GBT); s_ fB = findstr('MAXIMUM', GBT); cr = native2unicode(10); space = native2unicode(9); for j = 1:length(s_iB) BT = strtrim(GBT(s_iB(j)+4:s_fB(j)-1)); BT = strrep(BT, cr, space); BT = str2num(BT); beta(j) =BT(1,2); end string = ['arc',num2str(arclength),'_ asp',num2str(aspect),str1]; fi d2 = fopen(['C:\Documents and Settings\sjagirda\Directory200steps \',string,'\',string,'PHI.txt']); APH = fread(fid2);

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109 Appendix C (Continued) fclose(fid2); GPH = native2unicode(APH)'; s_iP = findstr('ROTY', GPH); s_ fP = findstr('MAXIMUM', GPH); cr = native2unicode(10); space = native2unicode(9); for j = 1:length(s_iP) PH = strtrim(GPH(s_iP(j)+4:s_fP(j)-1)); PH = strrep(PH, cr, space); PH = str2num(PH); PHI(j) =PH(1,2); end PHI(j); fi d3 = fopen(['C:\Documents and Settings\sjagirda\Directory200steps \',string,'\',string,'DISPX.txt']); DISPXA = fread(fid3); fclose(fid3); DI SPXG = native2unicode(DISPXA)'; s_iX = findstr('UX', DISPXG); s_fX = findstr('MAXIMUM', DISPXG);

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110 Appendix C (Continued) fi d4 = fopen(['C:\Documents and Settings\sjagirda\Directory200steps \',string,'\',string,'DISPY.txt']); DISPYA = fread(fid4); fclose(fid4); DI SPYG = native2unicode(DISPYA)'; s_iY = findstr('UY', DISPYG); s_fY = findstr('MAXIMUM', DISPYG); fi d5 = fopen(['C:\Documents and Settings\sjagirda\Directory200steps \',string,'\',string,'DISPZ.txt']); DISPZA = fread(fid5); fclose(fid5); DI SPZG = native2unicode(DISPZA)'; s_iZ = findstr('UZ', DISPZG); s_fZ = findstr('MAXIMUM', DISPZG); cr = native2unicode(10); space = native2unicode(9); for j = 1:length(s_iX)

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111 Appendix C (Continued) M = st rtrim(DISPXG(s_iX(j)+4:s_fX(j)-1)); M = strrep(M, cr, space); M = str2num(M); NODE12DISPX =M(1,2); NODE13DISPX =M(2,2); NODE14DISPX =M(3,2); NODE15DISPX =M(4,2); N = st rtrim(DISPYG(s_iY(j)+4:s_fY(j)-1)); N = strrep(N, cr, space); N = str2num(N); NODE12DISPY =N(1,2); NODE13DISPY =N(2,2); NODE14DISPY =N(3,2); NODE15DISPY =N(4,2); O = st rtrim(DISPZG(s_iZ(j)+4:s_fZ(j)-1)); O = strrep(O, cr, space); O = str2num(O); NODE12DISPZ =O(1,2); NODE13DISPZ =O(2,2);

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112 Appendix C (Continued) NODE14DISPZ =O(3,2); NODE15DISPZ =O(4,2); B = [ cos(be ta(j)) sin(beta(j)) 0 ; -sin(beta(j)) cos(beta(j)) 0 ; 0 0 1 ]; new cs = [ 50 0 0 ; 0 50 0 ; 0 0 -50 ]; dispatbeta=[ NO DE12DISPX NODE12DISPY NODE12DISPZ ; NODE13DISPX NODE13DISPY NODE13DISPZ ; NODE14DISPX NODE14DISPY NODE14DISPZ ; NODE15DISPX NODE15DISPY NODE15DISPZ ]; orgcoor d = [ 1000 0 0; 1050 0 0; 1000 50 0; 1000 0 -50]; Finalcoord=dispatbeta+orgcoord; node12=[ Finalcoord(1,1:3);Finalcoord(1,1:3) ;Finalcoord(1,1:3);Finalcoord(1,1: 3)]; positi on_vectorbeta=Finalcoord node12;

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113 Appendix C (Continued) position_vectorbeta(1,:) = []; A = B*newcs*inv(position_vectorbeta); thetaobeta(j)=acos(A(2,2)); end lambda=arclength*pi/180; BG=zeros(arclength,151); Beta=0; for countk=1:1:501 for countBETA=1:1:200 oldbeta=Beta; newbeta=beta(1,countBETA); if ne wbeta==oldbeta countBETA=countBETA-1; break; else Beta=newbeta; gamma=(countk/2000+.7495); gamma_l = gamma*lambda; phi=PHI(1,countBETA);

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114 Appendix C (Continued) sincapth eta=sin(Beta)./sin(gamma_l); coscaptheta=tan((lambda-phi)-(lambdagamma_l)).*cot(gamma_l); % captheta = atan(tan(Beta)./sin((lambda-phi)(lambda-gamma_l))); captheta = atan2(sincaptheta,coscaptheta); abs_epsilo n_e = sqrt((cos(Beta).*cos(phi)ones(size(Beta))).^2+(sin(Beta)).^2 .*(cos(phi)).^2+(sin(phi)).^2); epsilon _ex = cos(Beta).*cos(phi)-1; epsilon_ey = sin(Beta); eps ilon_ez = cos(Beta).*sin(phi); epsilo n_ax = (cos(gamma_l)).^2.*(1cos(captheta))+cos(captheta)-1; epsilon_ay = sin(captheta).*sin(gamma_l); epsilon_az = sin(gamma_l).*cos(gamma_l).*(1cos(captheta)); error = sqrt((epsilon_ex-epsilon_ax).^2 +(epsilon_ey-epsilon_ay).^2 +(epsilon_ez-epsilon_az).^2); rel _error = error./abs_epsilon_e;

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115 Appendix C (Continued) captheta 1(countBETA,countk)=captheta; abs _epsilon_e1(countBETA,countk) = abs_epsilon_e; epsilon_ex1( countBETA,countk) = epsilon_ex; epsilon_ey1( countBETA,countk) = epsilon_ey; epsilon_ez1( countBETA,countk) = epsilon_ez; epsilon_ax1( countBETA,countk) = epsilon_ax; epsilon_ay1( countBETA,countk) = epsilon_ay; epsilon_az1( countBETA,countk) = epsilon_az; error1(co untBETA,countk)=error; rel_erro r1(countBETA,countk)=rel_error; if rel_error <= 0.005 error1(countBETA,countk)=error; rel_e rror1(countBETA,countk)=rel_error; BG(co untBETA,countk)=countBETA*countk; betamax(countk) = Beta; maxf ortheta(countk,countBETA)=Beta; else break end

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116 Appendix C (Continued) end end end [y,i] = max(betamax); gammastar = (i/2000+.7495); gammastar_l=lambda*gammastar; [p,q]=max(maxfortheta); [r,s]=max(q); Beta=beta(1,s); phi=PHI(1,s); thetaostar=thetaobeta(1,s); sincap thetastar=sin(beta)./sin(gammastar_l); cosc apthetastar=tan((lambda-PHI)-(lambdagammastar_l)).*cot(gammastar_l); capthetastar = atan2(sincapthetastar,coscapthetastar); % captheta star =atan2(tan(beta),sin((lambda-PHI)(lambda-gammastar_l))); [p2,s2] = po lyfit(capthetastar(1:s),thetaobeta(1:s),1);

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117 Appendix C (Continued) CTHETAstar=p2(1); GAMMA _MATRIX(counter,countas)=gammastar; CAPTHETA_MATRIX(counter,countas)=capthetastar(s)*(180/pi); CTHETA _MATRIX(counter,countas)=CTHETAstar; end %plot([0.1:.1:1],GAMMA_MATRIX(arclength,:)); %drawnow end GAMMA_MATRIX; CTHETA_MATRIX; CAPTHETA_MATRIX; figure(1) plot([4:2:112],GAMMA_MATRIX) xlabel('arc length \lambda (degrees)') ylabel('Characteristic radius factor, \gamma') figure(2) plot([4:2:112],CTHETA_MATRIX) xlabel('arc length, \lambda (degrees)')

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118 Appendix C (Continued) ylabel('Parametric angle coefficient, C_\Theta') figure(3) plot([4:2:112],CAPTHETA_MATRIX) xlabel('arc length, \lambda (degrees)') ylabel('Parameterization limit, \Theta_{max} (degrees) for the parametric angle coefficient, C_\Theta_{max}')

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119 Appendix D Summary Table 1: Spherical PRBM ) cos (tan tan ) 1 (1 ) sin (sin sin1 0=C ] )] ( ) sin[( tan [ tan1 2 / 1 2 2 2} ] sin [cos sin ] 1 cos {[cos e 2 / 1 2 2 2} )] cos 1 ( cos [sin ] sin [sin ] cos ) cos 1 ( {[cos a 2 / 1 2 2 2} )] cos 1 ( cos sin sin [cos ] sin sin [sin )] 1 cos ) cos 1 ( (cos ) 1 cos {[(cos error 2 / 1 2 2 2 2 / 1 2 2 2} ] sin [cos sin ] 1 cos {[cos } )] cos 1 ( cos sin sin [cos ] sin sin [sin )] 1 cos ) cos 1 ( (cos ) 1 cos {[(cos eerror For small angles when the sphere has a very large radius the above formulas can be approximat ed to the planar case.

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120 Appendix D (Continued) Table 2: Planar PRBM )) cos 1 ( 1 ( l a sin l b 0= C ) 1 ( tan1 l a b 2 2) ( b a le 2 2) sin ( )] cos 1 ( [ l la 2 2] sin ) / [( )]} cos 1 ( 1 [ ) / {( l b l a error 2 2 2 2) / ( ) / 1 ( ] sin ) / [( )]} cos 1 ( 1 [ ) / {( l b l a l b l a errore