Citation

## Material Information

Title:
Moment-dependent pseudo-rigid-body models for beam deflection and stiffness kinematics and elasticity
Creator:
Espinosa, Diego Alejandro
Place of Publication:
[Tampa, Fla]
Publisher:
University of South Florida
Publication Date:
Language:
English

## Subjects

Subjects / Keywords:
Flexible mechanisms
Curved beam
Parametric model
Ortho-planar motion
Virtual work
Dissertations, Academic -- Mechanical Engineering -- Masters -- USF ( lcsh )
Genre:
non-fiction ( marcgt )

## Notes

Abstract:
ABSTRACT: This thesis introduces a novel parametric beam model for describing the kinematics and elastic properties of ortho-planar compliant Micro-Electro-Mechanical Systems (MEMS) with straight beams subject to specific buckling loads. Ortho-planar MEMS have the ability to achieve motion out the plane on which they were fabricated, characteristic that can be used to integrate optical devices such as variable optical attenuators and micro-mirrors. In addition, ortho-planar MEMS with large output forces and long strokes could be used to develop new applications such as tactile displays, active Braille, and actuation of micro-mirrors. In order to analyze the kinematics and elasticity of a curved beam contained in a Micro Helico-Kinematic Platform (MHKP) device, this thesis offers an improved model of straight and curved flexures under compressive loads. This model uses an approach similar to the one applied to develop a regular Pseudo-Rigid -Body Model but it differs in the definition of a key parameter, the characteristic radius factor, y, which is not a constant, but a function of the moment, Å·=y(M) . This approach allows for the Pseudo-Rigid-Body Model (PRBM) to describe the motion taken by the deflected beam precisely over a large range of motion. In developing the model, this thesis describes kinematic and elastic parameters such as the angle coefficient, Câ‚‰, the characteristic radius, yl, and the torque coefficient, T[subscript Î˜]. Furthermore, the torque coefficient is divided into two component functions, T[subscript f], and, T[subscript m], which can be used to find the working loads (force and moment) on the beam. The input displacement is the only needed state variable, object variables, which describe the beam, include the material modulus of elasticity, E, the moment of inertia, I, and its length, l.
Thesis:
Thesis (M.S.M.E.)--University of South Florida, 2009.
Bibliography:
Includes bibliographical references.
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Title from PDF of title page.
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Document formatted into pages; contains 155 pages.
Statement of Responsibility:
by Diego Alejandro Espinosa.

## Record Information

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University of South Florida
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All applicable rights reserved by the source institution and holding location.
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002029599 ( ALEPH )
436997200 ( OCLC )
E14-SFE0002943 ( USFLDC DOI )
e14.2943 ( USFLDC Handle )

## USFLDC Membership

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USF Electronic Theses and Dissertations

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Full Text
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ABSTRACT: This thesis introduces a novel parametric beam model for describing the kinematics and elastic properties of ortho-planar compliant Micro-Electro-Mechanical Systems (MEMS) with straight beams subject to specific buckling loads. Ortho-planar MEMS have the ability to achieve motion out the plane on which they were fabricated, characteristic that can be used to integrate optical devices such as variable optical attenuators and micro-mirrors. In addition, ortho-planar MEMS with large output forces and long strokes could be used to develop new applications such as tactile displays, active Braille, and actuation of micro-mirrors. In order to analyze the kinematics and elasticity of a curved beam contained in a Micro Helico-Kinematic Platform (MHKP) device, this thesis offers an improved model of straight and curved flexures under compressive loads. This model uses an approach similar to the one applied to develop a regular Pseudo-Rigid -Body Model but it differs in the definition of a key parameter, the characteristic radius factor, y, which is not a constant, but a function of the moment, =y(M) This approach allows for the Pseudo-Rigid-Body Model (PRBM) to describe the motion taken by the deflected beam precisely over a large range of motion. In developing the model, this thesis describes kinematic and elastic parameters such as the angle coefficient, C, the characteristic radius, yl, and the torque coefficient, T[subscript ]. Furthermore, the torque coefficient is divided into two component functions, T[subscript f], and, T[subscript m], which can be used to find the working loads (force and moment) on the beam. The input displacement is the only needed state variable, object variables, which describe the beam, include the material modulus of elasticity, E, the moment of inertia, I, and its length, l.
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Moment-Dependent Pseudo-Rigid-Body Models for Beam Deflection and Stiffness Kinematics and Elasticity By Diego Alejandro Espinosa 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. Nathan Crane, Ph.D. Rajiv Dubey, Ph.D. Kathryn J De Laurentis, Ph.D. Date of Approval: March 24, 2009 Keywords: Flexible mechanisms, curved beam, parametric model, ortho-pl anar motion, virtual work Copyright 2009, Diego Espinosa

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Dedication To my family and all of those who belie ved and supported me th roughout this journey

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Acknowledgements I would like to express my gratitude to all of those who helped me and supported me throughout my education. I want to thank my mother, my father, and all the members of my family, who always believed in me a nd gave me courage to succeed and finish all of my goals. I want to give special thanks to Dr. Craig Lusk, who has been not only my mentor, but my friend and advisor. He was with me every step of the way, guiding me, teaching me, and showing me the right path for a successful career. I have learned so much from him and I am sure that without his caring dedication, knowledgeable advice and unceasing patience this thesis would not be a reality. Thank you very much, Dr. Lusk. I also would like to thank my comm ittee members Dr. Rajiv Dubey, Dr. Nathan Crane, and Dr. De Laurentis, who took the ti me to review my thesis and gave me valuable advice to improve and complete my work. Finally, I would like to thank the Mechanical Engineering Department, the Colle ge of Engineering, and the University of South Florida for opening the door to me and giving me the opportunity to explore, discover and use the best of my potential. Thank you very much.

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i Table of Contents List of Tables ................................................................................................................ ..... iii List of Figures ............................................................................................................... .......v Abstract ...................................................................................................................... ....... vii Chapter 1. Introduction ....................................................................................................... .1 1.1 Objective ............................................................................................................3 1.2 Motivation ..........................................................................................................4 1.3 Scope ..................................................................................................................5 1.4 Contributions......................................................................................................6 1.5 Roadmap ............................................................................................................7 Chapter 2. Background .......................................................................................................8 2.1 Pseudo-Rigid-Body Model ................................................................................8 2.2 Kinematics of the Pseudo-Rigid-Body Model ...................................................9 2.3 Elasticity of the Pseudo-Rigid-Body Model ....................................................12 2.4 Trigonometric Relationships Between Planar and Spherical Mechanisms .....13 2.5 Closure .............................................................................................................18 Chapter 3. Methodology and Model Development............................................................19 3.1 Theoretical Approach of Novel Parame tric Beam Model for Straight and Curved Beams Kinematic Analysis .................................................................20 3.2 Computational Approach for Stra ight Compliant Beam Deflection ................26 3.3 Kinematic Analysis Results La rge Deflection Straight Beams .......................31 3.4 Computational Approach for Curv ed Compliant Beam Deflection ................41 3.5 Curved Beam Kinematic Analysis ...................................................................44 3.6 Kinematic Analysis Results Large Deflection Curved Beams ........................53 Chapter 4. Elasticity of a PRBM for Curved Beams .........................................................60 4.1 Principle of Virtual Work ................................................................................60 4.2 Model Validation .............................................................................................65 4.3 Analysis of a Compliant Micro Helico-Kinematic Platform (MHKP) Device ..............................................................................................................68 Chapter 5. Conclusion ........................................................................................................7 0 5.1 Conclusion and Summary of Contributions .....................................................70

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ii References .................................................................................................................... ......72 Appendices .................................................................................................................... .....75 Appendix A: ANSYS Batch Code for a Vertical End-Loaded Beam ...................76 Appendix B: MATLAB Code for a Vertical End-Loaded Beam ..........................88 Appendix C: ANSYS Batch Code for a Specific Horizontal Buckling End-Loaded Beam ...........................................................................93 Appendix D: MATLAB code for a Specific Horizontal Buckling End-Loaded Beam .........................................................................105 Appendix E: ANSYS Batch Code for a Specific Horizontal Buckling End-Loaded Curved Beam .............................................................110 Appendix F: MATLAB Code for a Specific Horizontal Buckling End-Loaded Curved Beam .............................................................130 Appendix G: Tables of Summary of Re sults for a Specific Horizontal Buckling End-Loaded Curved Beam .............................................152

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iii List of Tables Table 3.1 Notation of straight b eam and curved beam variables ...............................20 Table 3.2 Object data for FEA ...................................................................................27 Table 3.3 Values of the rational function coefficients for Case 1: Vertical end-load ...........................................................................31 Table 3.4 Results of statistic anal ysis for a vertical end load ...................................36 Table 3.5 Values of the rational function coefficients for Case 2: Horizontal buckling end-load ........................................................37 Table 3.6 Results of statistic analysis for a horizontal buckling end-load ................40 Table 3.7 Object data for FEA of curved beams ........................................................41 Table 3.8 Coordinate frames position and orientation ...............................................46 Table 3.9 Nomenclature .............................................................................................47 Table 3.10 Values of the rational fu nction coefficients for a curved beam with a horizontal buckling end-load .................................................54 Table 3.11 Results of statistic an alysis for a horizontal buckling end-load curved beam ................................................................................58 Table 4.1 Torque function coefficients ......................................................................64 Table 4.2 Characteristics of the test beam .................................................................67 Table 4.3 MHKP material properties and cross-sectional area characteristics ..........68 Table 4.4 MHKP device simulation results ...............................................................69 Table G.1 Summary of results of curved beam with ..................................152 Table G.2 Summary of results of curved beam with ..................................153

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iv Table G.3 Summary of results of curved beam with ..................................153 Table G.4 Summary of results of curved beam with ..................................154 Table G.5 Summary of results of curved beam with ..................................154 Table G.6 Summary of results of curved beam with ..................................155 Table G.7 Summary of results of curved beam with ................................155

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v List of Figures Figure 2.1 A compliant can tilever beam w ith beam tip at x=a and y=b .....................10 Figure 2.2 The PRBM of a compliant cantilever beam ...............................................10 Figure 2.3 Spherical triangle with sides k m and n ; and dihedral angles and ........................................................................15 Figure 2.4 NapierÂ’s Circle for a spheri cal right triangle with a right angle .............17 Figure 3.1 Shows the undeflected position of the beam Case 1 ..................................28 Figure 3.2 Shows the deflected position of beam Case 1 ............................................28 Figure 3.3 Shows the undeflected position of the beam Case 2 ..................................29 Figure 3.4 Shows the deflected position of beam Case 2 ............................................30 Figure 3.5 versus moment for Case 1: Vertical end-load .........................................32 Figure 3.6 Case 1: Vertical endload horizontal and vertical position of beam end ..................................................................................33 Figure 3.7 Case 1: Vert ical end-load moment versus ...........................................34 Figure 3.8 Shows the characteristic radius factor within the 95% confidence interval for Case 1: Vertical end load ......................................35 Figure 3.9 versus moment for Case 2: Horizontal buckling end-load .....................37 Figure 3.10 Case 2: Horizontal buckli ng end-load, horizonta l and vertical position of beam end ..................................................................................38 Figure 3.11 Case 2: Horizont al buckling end-load moment versus ........................39 Figure 3.12 Shows the characteristic radius factor within the 95% confidence interval for Case 2: Horizontal buckling end-load ..................40

PAGE 9

PAGE 10

vii Moment-Dependent Pseudo-Rigid-Body Models for Beam Deflection and Stiffness Diego Alejandro Espinosa ABSTRACT This thesis introduces a novel parame tric beam model for describing the kinematics and elastic properties of ortho-pl anar compliant Micro-Electro-Mechanical Systems (MEMS) with straight beams subjec t to specific buckling loads. Ortho-planar MEMS have the ability to achieve motion out the plane on which they were fabricated, characteristic that can be used to integrat e optical devices such as variable optical attenuators and micro-mirrors. In addition, ortho-planar MEMS with large output forces and long strokes could be used to develop ne w applications such as tactile displays, active Braille, and actuation of micro-mirrors In order to analyze the kinematics and elasticity of a curved beam contained in a Micro Helico-Kinematic Platform (MHKP) device, this thesis offers an improved m odel of straight and curved flexures under compressive loads. This model uses an appro ach similar to the one applied to develop a regular Pseudo-Rigid Â–Body Model but it differs in the definition of a key parameter, the characteristic radius factor , which is not a constant, bu t a function of the moment, M Âˆ. This approach allows for the Pseudo -Rigid-Body Model (PRBM) to describe the motion taken by the deflected beam pr ecisely over a large range of motion. In developing the model, this thesis describes ki nematic and elastic para meters such as the

PAGE 11

viii angle coefficient C the characteristic radius l and the torque coefficient T. Furthermore, the torque coefficient is divided into two component functions, Tf, and, Tm, which can be used to find the working load s (force and moment) on the beam. The input displacement is the only needed state variable, object variables, which describe the beam, include the material modulus of elasticity, E the moment of inertia, I and its length, l

PAGE 12

1 Chapter 1 Introduction Mechanisms are Â“mechanical devices use to transfer or transform energy, force, or motion [1, 2]. We can distinguish between rigid-body mechanisms, which are Â“systems of rigid links connected by movable jointsÂ” such as cams, lever, and gears and compliant mechanisms which Â“gain some of their mobility from the deflection of flexible members rather than from movable joints onlyÂ” [3]. Compliant mechanisms offer advantages such as cost reduction and perf ormance improvements, which are achieved by increasing the precision and reliability of the mechanism, reducing wear, weight, maintenance, assembly time and the number of parts needed for assembly [3]. These advantages are particularly im portant in devices fabricated at the micro scale, known as Micro-Electro-Mechanical Systems (MEMS). MEMS technology offers ways to bridge fi elds that were previously unrelated, joining different branches of study such as biology and microelectr onics. Some examples of the applications in which MEMS are cu rrently being applied include biotechnology, where MEMS have facilitated new discoveri es Â“such as Polymerase Chain Reaction (PCR) Microsystems for DNA amplificati on and identification, micro-machined Scanning Tunneling Microscopes (STMs), biochi ps for detection of hazardous chemical and biological agents and micro-systems for drug screen ing and selectionÂ” [4]. MEMS

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2 technology has also improved the overall perf ormance of communicatio n circuits and has reduced the cost and power consumpti on of such devices. Moreover, MEMS accelerometers are rapidly substituting for the conventional accelerometers used to trigger air-bag deployment systems in automobile s, due to their reliab ility, the ability to integrate the accelerometer and electronics in a single silicon chip, and reduction of the overall cost [4]. Generally MEMS are fabricated by depositing multiple planar layers of polysilicon, or polycrystalline silicon on a silicon wafer, then, by bulk micromachining, planar lithography is used to se lectively etch and shaped the planar material layers into the micro-structure desired [3]. There are se veral challenges in the design phase and part assembly of mechanical devices at the micr o level. Due to the planar nature of the fabrication process and the small scale in wh ich MEMS are designed, it is hard to create hinges and pin joints that accurately move a nd stay in place. Additionally, it is also challenging to design three-dimensional motion devices that achieve their specific tasks without failing first. The performance of these devices greatly depend on the materials properties, yet the material choices for ME MS processes are very limited and their behavior and properties at the micro-s cale are not completely understood [3]. Nevertheless, compliant mechanisms offer an answer to many of these problems due to advantages they offer in the designs of mech anisms at the micro level [5]. According to Clements [5] Â“Compliant MEMS: Can be fabricated in a plane Required no assembly Required less space and are less complex

PAGE 14

3 Have less need for lubrication Have reduced friction and wear Have less clearances due to pin join ts, resulting in higher precision Integrate energy storage elements (s prings) with the other componentsÂ” One of the methods used to performed co mpliant beams deflection analysis is a mathematical approach based on elliptical inte grals, a method that is difficult to use and does not produce much insight on the mo tion and the stiffness of the beam. Consequently, alternative methods have been found in order to make the analysis simpler and more intuitive. One of these methods is a parametric approxi mation model called the Pseudo-Rigid-Body Model (PRBM) in which the compliant mechanism is modeled by an analogue rigid mechanism [3 ]. Some of the weaknesses of this model are: First, it only works on certain load and force configurations. Second, for some long and thin beams it does not model the compliant beam deflection throughout its co mplete range of motion. Finally, it does not work on curved beams. 1.1 Objective The objectives of this thesis are: To create a more accurate parametric b eam model for compliant MEMS using a rational function to represent the characteris tic radius factor as a function of the moment load, ÂˆM) Second, apply this model for the analysis of the kinematics and elasticity of the complete deflection range of motion of both straight and the curved beams.

PAGE 15

4 Third, find specific buckling loads on a compliant Micro Helico-Kinematic Platform (MHKP) device [6]. Finally, develop software codes in orde r to produce a new parametric model and provide validation of its cap abilities see appendixes A, C, and E (ANSYS codes), B, D, and F (MATLAB codes). 1.2 Motivation Ortho-planar mechanisms are a subclass of mechanisms that are manufactured in a single plane and have the ability to achieve motion out of that plan e [7]. Ortho-planar MEMS can be used to Â“integrated optical devi ces such as variable optical attenuators, micro Fresnel lenses, micro grating, multiplexers, and micro-mirrors Â” [8]. In addition, ortho-planar MEMS with a high integration ability and capability of moving in the outplane direction with large out put forces and long strokes c ould be used to develop new applications such as tactile displays, active Braille, and actuation of micro-mirrors [9]. Last but not least, ortho-planar MEMS coul d be implemented to developed new spatial light modulators [10] capable of manipulating optical wave fronts, which can be Â“used in image projectors, optical switchi ng, and adaptive opticsÂ” [11]. Planar kinematics has been the method of choice when designing ortho-planar MEMS, leaving the benefits of spherical kinematics larg ely unnoticed and unexplored. Â“Using spherical kinematics, it is possible to design devices th at move links to specified spatial orientations and can c overt rotation in one plane to rotation in a different planeÂ” [11]. The intent of this work is to provide tools that can help design and analyze curved

PAGE 16

5 beams, furthering and facilitating the expl oration and development of new compliant ortho-planar MEMS using spherical kinematics. 1.3 Scope The Micro Helico-Kinematic Platform (MHKP) is an example of an ortho-planar mechanism. It was designed using spheri cal kinematic techniques in which three spherical crank-sliders with th e same center and with an in-p lane rotational input are used to vertically translate (in the out-plane direc tion) and rotate a platform. Due to the nature of the input, this device does not produce si de-to-side motion, which allows for a closed packed array of similar devices and provides large vertical tran slation of the platform [6], making it a good candidate for movable pixe ls mirror [8]. To improve the design, manufacture process, and overall performance of this device, the rigid links, movable pins joints, and hinges were replaced by their compliant analogues; however, the kinematic and elastic analysis of the compliant analogues is more challenging due to the absence of a spherical Pseudo-Rigid-Body M odel for curved compliant beams loaded with buckling loads [12]. As a consequence, there was a need for a new spherical PseudoRigid-Body Model that would allow us to accu rately analyze the kinematic and elastic behavior of the MKKP curved compliant beams. This thesis introduces a novel parametric model for the kinematics and the elastic analysis of curved compliant beams using specific buckling load s. It focuses on the development of model parameters needed to ac curately describe the behavior of a curved beam as it is deflects. Consequently, one can use this new model to accurate describe the

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6 kinematics of the deflected MHKPÂ’s curved beams and computed the loads needed to actuate the device. 1.4 Contributions The major contributions made by this resear ch work in the analysis and design of compliant beams include: A novel parametric beam model for the ki nematic analysis of buckled beams, where characteristic radius factor, is a function of the moment load, ÂˆM) Improvements in accuracy and range of this model compared with previous models by using a rational function model. The development of non-dimensional kinema tic and elastic parameters, the angle coefficient, C the characteristic radius factor, and the torque coefficients, T, Tf, and, Tm, that when combined with object data, the modulus of elasticity, E the moment of inertia, I the length of the beam, l and state information, such as the input displacement, d or the input rotation, can be use to determine the buckling behavior of the beam. This thesis also describes finite elemen t analysis (FEA) procedures that were used to develop and validate the new parametric beam model. This thesis computes specific buckling lo ads needed to actuate a compliant Micro Helico-Kinematic Platform (MHKP) device. This thesis establishes the coordinates of the highest point reached by the MHKP once it is buckled.

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7 Finally, this work provide s MATLAB codes used to analyze the output data produced by the finite element analysis, derive and validate the new parametric model, and obtain the kinema tic and elastic parameters. 1.5 Roadmap This first chapter has introduced the work developed in this thesis. Chapter 2 provides the background of PRBMs and spheri cal kinematics. Chapter 3 explains the relationships between a planar PRBM and a spherical PRBM. It also describes the approach taken to develop the new parametr ic beam model and its outcomes. Chapter 4 gives an overview of the prin ciple of virtual work and how it was used to develop the torque coefficient, T, and its components, Tf, and, Tm. Finally, Chapter 5 discusses the results and validation and provides the conclusions.

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8 Chapter 2 Background This chapter reviews previous work on Pseudo-Rigid-Body Models and spherical trigonometry. These are explained in detail b ecause they are the starting point for this work and their overall unders tanding is essential for an easier comprehension of the details in this thesis. 2.1 Pseudo-Rigid-Body Model In the past, elliptic integrals have been used to analyze end-loaded large deflection cantilever beams in order to obtain closed-form solutions [3]. However, this mathematical approach is difficult to use and produces little insight about the motion or stiffness of the beam. As a result, alternativ e methods of determining the beam deflection path have been developed, one of these is a parametric approximation model called the Pseudo-Rigid-Body Model (PRBM). This met hod consists in describing the compliant memberÂ’s motion and stiffness by replacing it with a rigid-link analogue that has approximately the Â“same motion and stiffn ess for a known range of motion and to a known mathematical toleranceÂ” [13]. In othe r words, the PRBM Â“provides a simplified but accurate method of analyzing the deflec tion of flexible beams and provides the designer a means of visu alizing the deflectionÂ” [3], mean ing that given a compliant beam,

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9 its motion can be found by treating it as a mech anism with rigid links or given a specific motion, a PRBM that performs the same motio n can be developed and transformed into its compliant analogue [13]. After the iden tification of the PRBM of the compliant member, its kinematic and elastic parameters ar e optimized and validated so that its range of applicability and level of error are known and acceptable [13]. 2.2 Kinematics of the Pseudo-Rigid-Body Model This PRBM approach is based on the fact that the deflection of the beamÂ’s free end follows a near-circular path with a cente r of curvature located at a point on the undeflected beam [3]. This allows the PRBM to determine Â“the relativ e positions of the end points of various compliant segments wit hout precise modeling of the locations of interior points [8]. In a ddition, the PRBM is used to compute the amount of force required to achieve the desired deflection. An example of a PRBM for a straight cantilever beam with a vertical end load [3] is s hown in Figure 2.1. This model was developed by Howell et al. [3,14] and is show n here to provide context for the current research.

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10 Figure 2.1. A compliant cantilever beam with beam tip at x= a and y=b. Adapted from [3] Figure 2.2. The PRBM of a compliant cantileve r beam. (See Figure 2.1) adapted from [3]

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11 Figure 2.1 shows the compliant cantilever beam and the large-deflection path of the beam end and Figure 2.2 shows a PseudoRigid-Body Model that approximates the nearly circular path of the beam end. The Pseudo-Rigid-Body Model is created with two rigid links joined at a poi nt along the beam called the characteristic pivot The location of the characteristic pivot is chosen so that th e path of the beam end of the rigid model matches, as closely as possible, the path of the beam end of the compliant beam. The distance from the beam end to the characteristic pivot is called the characteristic radius l where the constant, is named the characteristic radius factor The angle known as the Pseudo-Rigid-Body angle is the amount of ro tation that the rigid link must undergo to match the deflection of the compliant beam Furthermore, the angl e of inclination of the compliant beam at the beam end is given by0 In addition, the horizontal ( x coordinate) and the vertical ( y -coordinate) coordinates of th e end of the deflected beam are represented by the variables a and b respectively, which are given in terms of the PRBM angle, in equations (2.1) and (2.2) Thus, the value of can be calculated using equation (2.3). The relation between and 0 is represented by (2.4), where C represents the angle coefficient with a value of 1.24. cos 1 1l a (2.1) sinl b (2.2) 1 tan1l a b (2.3) C0 (2.4)

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12 where Crepresents the angle coefficient with a value of 1.24. In a research monograph by Howell [14], the charact eristic radius factor, was found by establishing the maximum acceptable percent error in the deflection at 0.5% equation (2.5); then, optimization was used to determine the value of that allows for a maximum value of PRBM angle, without violating the maxi mum error constraints. It was determined that an optimal value of is 0.8517, which maintains an error smaller than 0.5% at angular deflections less than max=64.3. max max0 for error error ge e (2.5) Where eerrorrepresents the error of the rela tive deflection between the PRBM and the compliant member and e represents is the vector difference of the deflected position of the beams end point and its undeflected position. 2.3 Elasticity of the Pseudo-Rigid-Body Model In order to model the elasti city of the material and its resistance to deformation, a torsional spring with a constant spring-rate, K is placed at the characteristic pivot, as shown in Figure 2.2. When a load is a pplied to a PRBM link at an angle, the component of the force orthogonal to the beam Â’s surface and tangent to the end pointÂ’s path is represented by Ft, which is defined by equation (2.6) [3]. l K F Ft sin (2.6)

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13 This transverse force, Ft, is responsible for the initia l deflection of the rigid link, and creates a torque, T about the characteristic pivot. l F Tt (2.7) Substituting (2.6) in (2.7) yields the torque required to deflect the flexural beam in terms of the constant spring-rate, K and the PRBM angle, K T (2.8) The value of the torsional sp ring constant spring-rate, K can be calculated as a function of the geometry of the beam, I/l its material properties, E the PRBM constants, ,and the nondimensionalized spring constant, K, defined as the stiffness coefficient equation (2.9). l EI K K (2.9) K allows for an easy calculation of th e force necessary to deflect the PseudoRigid-Body Model; this force is approximately equal to the force required to deflect the compliant member equation (2.10), however, th e elastic portion of the PRBM yields a max ( K)<58.5 in order to have an accu rate force prediction [15]. 2l EIK Ft (2.10) 2.4 Trigonometric Relationships Between Planar and Spherical Mechanisms This section reviews the basi cs of spherical trigonometry and introduces NapierÂ’s rules, which later are used to analyze spherical triangles.

PAGE 25

14 Planar mechanisms are those in which jo ints axes are parallel and links are defined by the lengths between joints; in co ntrast, spherical mechanisms are those in which the joints axes intersect at the cente r of a sphere and links are defined by their great circles arcs [16]. A great ci rcle is defined as one that has a radius equal to the radius of the sphere, and it is contained within a pl ane that intersects the sphere. For example, the earth longitudinal circles are great circles; the only latitudinal circle that is a great circle is the equator, which shares the same radius as the earth. Moreover, the angles between the planes containing the great ci rcles are defined as dihedral angles. Relationships between planar and spherical c onfigurations can be developed [17], often allowing the application of familiar planar kinematics concepts to spherical configurations. In spherical trigonometry the surface is not flat; instead, it is the curved surface of a sphere, where neither straight lin es nor planar figures can be drawn [8]. However, Â“there are geometrical features on a spherical surface that have properties mathematically similar to their planar counterpa rtsÂ” [8]. For instance, a circle arcs drawn on a surface of a sphere possesses similar mathematical properties as a straight line drawn on a plane; furthermore, angles between inte rsecting circles can become analogues to the angles formed by straight lines [18]. These types of similari ties allow the application of similar relationships and trigonometric rules, su ch as the Law of Sines, Law of Cosines, and rules for spherical right triangles, (Spheri cal triangle in which one of the dihedral angles is 90). The following procedure follows Spiege l and LiuÂ’s work [17] to describe the analogies between spherical trigonom etry and plane trigonometry. During the discussion of these laws it is important to follow the nomenclature and differentiate between dihedral angels denoted by Greek letters, segment of great circles (arcs) denoted

PAGE 26

15 by lower-case roman letter, and the points on great circles denoted by upper-case roman letters. The measurement of the dihedral an gles and the great circles segments can be given in degrees or radians in order to facili tate the description of the spherical analogues of the Law of Sines and the Law of Cosine s, simplified relations for spherical right triangles, and allows for the de rivation of general results, which are independent of the radius of a particular sphere. A spherical triangle is show n in Figure 2.3, where the arcs and the dihedral angles can be related by the Laws of Sines and Cosines. Figure 2.3. Spherical triangle with sides k m and n ; and dihedral angles and Adapted from [8] The spherical Law of Cosines can relate the ar cs and the dihedral angles in the following ways:

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16 Three arcs and one dihedral angle sin sin sin cos cos cos n m n m (2.11) Three dihedral angles and one arc k sin sin sin cos cos cos (2.12) The spherical Law of Sines can relate two arcs and their respec tive opposing dihedral angles. sin sin sin sin sin sin k m n (2.13) A spherical right triangle is one whit at least one 90 dihedral angle, two other dihedral angles, and and three arcs, k m and n where n represents the hypotenuse of the spherical right triangle. If any two of these five parameters are known, one can find a third using NapierÂ’s Rules in spherical trigonometry. NapierÂ’s Rules are based on Figure 2.4, which arranges all five parame ters around a circle. The hypotenuse and the two non-right dihedral angles carry the prefix Â“coÂ”, which represents the complement of the angle. Any segment of the circle can be consider as a middle part the adjoining segments, are defined as the adjacent parts and the two remaining segments are defined as the opposite parts.

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17 Figure 2.4. NapierÂ’s Circle for a spherica l right triangle w ith a right angle Adapted from [17] NapierÂ’s Rules summarizes ten equations that describe the re lationships between any set of three of the five parameters in the NapierÂ’s Circle. The rules are: Rule 1: The sine of any middle part equals the product of the tangents of the adjacent parts. Rule 2: The sine of any middle part e quals the product of the cosines of the opposite parts. For instance, using Napie rÂ’s Rules the value of k gives for Rule 1: m m co k tan cot tan tan sin (2.14) and for Rule 2: sin sin cos cos sin n co n co k (2.15)

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18 2.5 Closure This chapter has reviewed previous work on the Pseudo-Rigid-Body Model developed by Howell and the basics of spheri cal trigonometry. In the next chapter those concepts will used to develop new parametr ic beam models for straight and curved beams.

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19 Chapter 3 Methodology and Model Development This section explains the theoretical approach taken for the straight beams deflection kinematic analysis. The notation of the variables of cu rved beam variables change due to the spherical nature of the beam and the need to expresses many of the variables as angles. However, there is a correspondence principl e between planar and spherical PRBMs, which states that Â“when sm all angle assumption is used for spherical arcs i.e. the arc length is much smaller than the radius of the sphere, the spherical PRBM become identical to a planar PRBM.Â” [12]. For instance, the arc length, can be express to the planar length, b Thus by small angle assumption: 1 cos (3.1) b sin (3.2) In the same way, the planar equivalents of and are a and l respectively. Additionally, the terminology used to represent angles and ratios such as , in s pherical and planar PRBMs is the same for both cases. See Table 3.1. The theoretical approach exposed in s ection 3.2 is analogues to the one of the curved beam. For practical reasons the e quations on this section are only express on terms of the straight beam variables. The curv ed beam kinematic analysis will be study in more depth in sections 3.4, 3.5, and 3.6.

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20 Table 3.1. Notation of straight beam and curved beam variables. Parameter Straight beam Curved Beam Characteristic radius factor Length/Arc-length l R End-beam x-coordinate b End-beam y-coordinate a Pseudo-Rigid-Body angle 3.1 Theoretical Approach of Novel Parame tric Beam Model for Straight and Curved Beams Kinematic Analysis In this work, a different approach is implemented for finding the characteristic radius factor, of straight and curved compliant b eams with large deflections. It is possible to regard equations (2.1) and (2.2) as definitions of and rather than a convenient approximation. Thus, given the coordinates of the beam end, a and b, for a particular load condi tion, the values of and can be found that satisfy equations (2.1) and (2.2) for that load. By squaring equations (2.1) and (2.2), (3.3) and (3.4) are defined in terms of a and b as: 2 2 2 2 *2 2 2 l a l b l al a (3.3) 1 *1 tanl a b (3.4) The crucial difference between this a pproach and HowellÂ’s method is that is not a constant value throughout the deflection motion. Rather, it is a function of the load that allows for the PRBM to describe a closer approximation of the motion taken by the deflected beam. As a result, it is required to find a function that can closely match the

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21 behavior of as the beam deflects and will not add significantly more complexity to the analysis of the PRBM. Our solution to that problem is to implement a rational function model that will fit the behavior of *. This type of function was chosen over a polynomial function because of its numerous mathemati cal advantages, some of which include, the qualitative superiority of a rational func tion over a polynomial function [19], and its ability to fit complex shapes while keepi ng a low degree in both the numerator and denominator. This means that the function wi ll required fewer coefficients giving it a relative simple form. Some other advantag es of rational functions [20] include: Excellent interpolatory propert ies and extrapolatory powers. The tendency to be smoother and not as oscillatory as a polynomial functions. Accurate asymptotic propert ies, they can model a function not only within the domain of the data but also to represent theoretical/asymptotic behavior outside the domain of interest. It is possible to crea ted an expression of M) which contains both positive and negative powers of M. For example, we can approximate: 2 2 1 0 1 2 21 Âˆ M b M b b M b M b M M (3.5) Where, iM= ith value of nondimensionalized M from data i= value of calculated form ai and bi from equation (3.1) i Âˆ= iM Âˆ= Rational fit to i In order to find the coefficients ( bi)

PAGE 33

22 Let, Tb b b b b B2 1 0 1 2 (3.6) B Xi Âˆ (3.7) Where, 2 2 2 2 2 1 1 1 2 11 1 1 1 1 1 1 1 1 1 1n n n n i i i iM M M M M M M M M M M M X (3.8) Using the method of least squares to find B 1 i T TX X X B (3.9) Once ib is known, equation (3.5) can be used to find MB Âˆ Then, the Pseudo-RigidBody angle, B can be found. 2 2 1 2 0 1 2Âˆ M b M b M b M b b MB (3.10) M l a bB BÂˆ 1 tan1 (3.11) In order to determine how well the parametric model, MB Âˆ, truly represents the data, statistical analysis a two-sided t test with a 95% confidence interval and the coefficient of determination 2 BR were performed. The 2 BR is used to determine a proportion of the variance of one variable that is predictable from another variable; in addition, its measurement allows determina tion of how certain one can be in making

PAGE 34

23 predictions based on certain model/graph. In othe r words, it represents the percent of the data that is closest to the line of best fit. Th e coefficient of determination is defined as the ratio of the explained variation to the total va riation [21, 22]. 2 *) ( Âˆ MB B (3.12) 2 * i iSST (3.13) 2 i BSSE (3.14) s Po data of Number SSEBint2 (3.15) SST SSE RB B 12 (3.16) Where2 B, represents the variance, B ,represent the squares explained, SSE represents the sum of the squares explai ned (explained variance), and SST represents the total sum of the squares (total variance). Then one can find the covariance matrix using, 2 1 B TX X VarB (3.17) However, computation of the variance of the coefficients of B is complicated because VarB is not a diagonal matrix (i.e. the basis functions chosen for X are not orthogonal). We can find an orthogonal set of basis functions by finding the eigenvectors of1 X XT, which is a symmetric matrix and has eigenvalu es that are real and its eigenvectors are orthogonal. We express the eigenvectors as an ort hogonal (rotation) matrix of eigenvectors and the eigenvalues as a diagonal matrix such that: VD V X XT 1 (3.18)

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24 Where, V = Eigenvectors matrix D = Diagonal Eigenvalues matrix We then can change the set of basis functions from the non-orthogonal set in X to an orthogonal set in X2, where V X X 2 (3.19) We can find the coefficients in the orthogonal basis by computing 2 1 2 2 i T TX X X C (3.20) Then, the parametric modelÂ’s function for th e characteristic radius factor and the PRBM angle functions becomes 2Âˆ X C MC (3.21) M l a bC CÂˆ 1 tan1 (3.22) The statistical two-side t test and the exam ination of the coefficient of determination2 CRcan now be performed using the new variables 2 *) ( Âˆ MC C (3.23) 2 * i iSST (3.24) 2 i CSSE (3.25) values data of Number SSEC C2 (3.26) SST SSE RC C 12 (3.27)

PAGE 36

25 It can be shown that B V CT (3.28) and that: 2 2 B C because 2X is a orthogonal transf ormation (rotation) of X and det ( V )=1. Therefore, the variance of C is: 2 1 2 2 C TX X VarC (3.29) or 2 1 CD VarC (3.30) because VarC is a diagonal matrix, the standard deviation of i th component of C can be found as: ii iVarC C (3.31) or i C iC (3.32) Where the standard deviation of the i th component of C is the square root of the matrix element in the i th row and the i th column of VarC ii i CVarC (3.33) from equation (3.25), it can th en be shown that variance, 2 ,i B and the standard deviation i B, of the i th element of B are related to the standard deviation, Cj of the coefficients jC of the orthogonal basis the elements by: 2 2 Cj ij i BV (3.34)

PAGE 37

PAGE 38

27 The geometry and the material properties of the beams were defined as follows. (See appendixes A and C). Table 3.2. Object data for FEA. Case 1: Vertical end-load Case 2: Horizontal end-load Length, l 25 m 100 m Width, w 10 m 10 m Thickness, t 1 m 1 m Modulus, E 169 GPa 169 GPa The reason why there is a difference betw een the lengths of the two cases is because the beam in Case 2 is being buckled and just a fourth of the beam is analyzed this will be explained in further detail later in this chapter. The ANSYS 3D beam element beam4 was used on these models; in order to take into account the complete 3D flexibility of the beams. In other words, the bending deformations, axial deformations, and tors ional deformations occurring on the two principal bending planes were allowed for these analyses. For Case 1 the boundary conditions defining this model were specified as follows. Node1 at the beginning of the beam was constrai ned in all directions so it remained fixed throughout the deflection motion. A vertical di splacement was applied to node 5, rotation about the y -axis and translation on the x and z -directions were allowed, all other degrees of freedom were constrained. Moreover, th e remaining nodes on the beam were left unconstrained allowing the observation and an alysis of the beamÂ’s motion as it was deflected see Figures 3.1 and 3.2.

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28 Figure 3.1. Shows the undeflected position of the beam Case 1. Figure 3.2. Shows the deflected position of beam Case 1.

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29 For Case 2, the boundary conditions were id entical to those of Case 1 with the exception that node 5 at the end of the beam was allow to move in x -direction and all other translations and rotations were prevented, resulting in the motion shown in Figures 3.3 and 3.4. Figure 3.3. Shows the undeflected position of the beam Case 2.

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30 Figure 3.4. Shows the deflected position of beam Case 2. The different loading conditions cases were defined as follows. Case 1 includes a vertical displacement load applied perpendicularly to the beamÂ’s end. Case 2 includes a buckling horizontal displacement applied horizontally to the beamÂ’s end as well. In both cases the beamÂ’s moving end was represente d by node 5, which displacement caused the translation and rotation of the rest of the nodes, the deflection of the beam, and reaction forces to take place at node 1 and node 5. Th is information was collected over a range of 300 load steps in an ANSYS data output file. Th is file contains the displacements of the nodes in the x y and z directions, their rotations about the x y and z axes, their reaction forces in the x y and z directions, and their re action moments about the x y and z axes as the beams were deflected. Once the ANSYS information was acquired, an analysis program was written in MATLAB; this program used the output info rmation file from ANSYYS to develop the

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31 new parametric beam model introduced in Section 3.2 (see appendixes B and D). This part of the model predicts the kinematic behavior of the deflected beams. 3.3 Kinematic Analysis Results Larg e Deflection Straight Beams This section provides a summ ary of the kinematic analys is of the deflection of a straight beam under differ ent loading conditions. For a vertical end-loaded beam Case 1, HowellÂ’s model, which uses a constant characteristic radius factor,, is accurate when the force acting on the beam is perpendicular to its neutral axis. However, as the end of the beam deflects and rotates, more of the beam becomes parallel to the di rection of the applied load, and the accuracy of the values predicted by his PRBM decrea ses dramatically. In other words, a PRBM that uses a constant works well for the initial part of the deflection and its accuracy diminishes as the beam undergoes larger defl ections as shown in Figure 3.5. On the other hand, in our new model, which uses a characte ristic radius factor defined as a rational function of the moment, M Âˆ Âˆ ,with coefficient constants bi, Table 3.3 the predicted values remain accurate throughout most of the deflection range. Table 3.3. Values of the rational function co efficients for Case 1: Vertical endload. Rational function Coefficients b-2 8.2e-1 b-1 1.0e-4 b0 -1.0e-5 b1 4.1e-1 b2 -1.9e-1

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32 Figure 3.5. versus moment for Case 1: Vertical end-load. Figure 3.6 shows the different PRBM appr oximations of the end-beam deflection path. As the beamÂ’s vertical deflection becomes large, significant portions of it become parallel to the applied force, this causes th e beam to elongate as is shown in the final vertical height being larger than the original length of the beam. Consequently, the PRBM using the constant does not describe the complete deflection motion nor the elongation of the member; in contrast, the new parametric model approximation is able to accurately describe the complete path of the end-beam and its elastic deformation.

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33 Figure 3.6. Case 1: Vertical end-load horiz ontal and vertical position of beam end. Figure 3.7 shows the approximation of the PRBM angle, found with the different parametric models. During the ini tial deflection motion of the beam, the PRBM using a constant exhibits an accurate prediction of the PRBM angle; however, as the deflection progresses the accuracy of this mode l diminishes giving a larger percent error for the prediction. Conversely, th e new parametric model shows a constant and accurate prediction of the PRBM a ngle throughout the complete deflection of the beam.

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34 Figure 3.7. Case 1: Vert ical end-load moment versus A statistical analysis was performed to determine how well the new parametric model predicts the kinematics of the beam. Co nsequently, a two-side d t test with a 95% confidence interval was devel oped; Figure 3.8 shows the char acteristic radius factor within the 95% confidence interval. The coefficient of determination 2 BRwas calculated to measure the amount of certainty of the mode l prediction, the rational function prediction maximum error and total error, and the constant g prediction total error in the deflection approximation were also calcula ted as shown in Table 3.4. The 95% confidence interval produces an upper and lower limit for each coefficient of the rational function used in the PRBM to model the beam. This interval indicates how much uncertainty there is in the estimate of the mean; the narrower the interval, the more precise is the estim ate. The coefficient of determination 2 BR represents

PAGE 46

35 the percentage of data that is closest to th e line of best fit descri bing how certain one can be in making a prediction using our model. Fina lly, the total error represents the error in the kinematic prediction of our model compared to the actual data provided by the FEA analysis. Figure 3.8. Shows the characteristic radius fact or within the 95% confidence interval for Case 1: Vertical end load.

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36 Table 3.4. Results of statistic anal ysis for a vertical end load. Rational function Coefficients 95% Confidence Interval b-2 8.2e-1 3.9e-3 b-1 1.0e-4 5.0e-4 b0 -1.0e-5 1.0e-5 b1 4.1e-1 1.2e-2 b2 -1.9e-1 1.2e-2 Coefficient of Determination, 2 BR 99.48% Rational Function Prediction Maximum Error 0.1495e-1 % Rational Function Prediction Total Error 2.7994e-3 % Constant Total Error 0.3160 % In Case 2, where a straight beam was lo aded with specific horizontal buckling loads, just the first fourth se gment of the beam was analyzed, that is from node 1 to node 2 as shown in Figures 3.3 and 3.4. This appr oach describes a cantilever beam loaded horizontally and at the same time permits the description of the buckled beam using symmetry. After testing with a constantPRBM it was noticed that a constant does not produce an accurate PRBM for the initial part of the beam deflection. Interestingly, the constant PRBM approximation improves when the beam deflects sufficiently, so that the force is more perpendicular to the neut ral axis of the beam. Furthermore, when is represented as a function of the load, i.e. as M Âˆ Âˆ a more accurate PRBM can be achieved throughout the whole range of the deflection. That is, regardless of the direction of the load or if the beam is being eval uated at the beginning or at the end of its deflection, M Âˆ Âˆ with coefficient constants bi Table 3.5, can always produce an accurate PRBM for the prediction of the beam behavior Figure 3.9.

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37 Table 3.5. Values of the rational function coefficients for Case 2: Horizontal buckling end-load. Rational function Coefficients b-2 8.216e-1 b-1 -1.27e-2 b0 -1.0e-4 b1 -1.72e-2 b2 -6.0e-3 Figure 3.9. versus moment for Case 2: Horizontal buckling end-load. Figure 3.10 shows the parametric approximati ons of the end-beam deflection path of the beam loaded with speci fic buckling loads. The new mo del is able to improve the accuracy of the PRBM in comparison with a constant model.

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38 Figure 3.10. Case 2: Horizontal buckling endload, horizontal and vertical position of beam end. Figure 3.11 shows the approximation of th e deflected member PRBM angle, Once again, the new parametric model dem onstrates more accurate large deflection kinematic model predictions than a PRBM using the constant

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39 Figure 3.11. Case 2: Horizontal buckling end-load moment versus For Case 2 a statistical analysis was perf ormed as well; a two-sided t test with a 95% confidence interval was developed; Fi gure 3.12 shows the char acteristic radius factor within the 95% c onfidence interval. The coefficient of determination2 BR, the rational function prediction maximum error and total er ror, and the constant prediction total error in the deflection approximation were also calculated as shown in Table 3.6.

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40 Figure 3.12. Shows the characteristic radius fact or within the 95% confidence interval for Case 2: Horizontal buckling end-load. Table 3.6. Results of statistic analysis for a horizontal buckling end-load. Rational function Coefficients 95% Confidence Interval b-2 8.216e-1 3.0e-43 b-1 -1.27e-2 1.0e-5 b0 1.0e-4 1.0e-5 b1 -1.72e-2 2.5e-3 b2 6.0e-31 2.4e-3 Coefficient of Determination, 2 BR 99.96% Rational Function Prediction Maximum Error 0.1704e-5 % Rational Function Prediction Total Error 1.6502e-7% Constant Total Error 0.3187 % A PRBM for a beam with specific horizo ntal buckling end-loads has only been previously published for a beam with initia l curvature. This is because a constant which was used for previous PRBMs does not yield accurate results. Consequently,

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41 taking this new approach, we can improve the accuracy of the PRBMs for the analysis of end-loaded cantileve r beams undergoing large deflections [24]. 3.4 Computational Approach for Curved Compliant Beam Deflection A model of a curved beam was developed in ANSYS in order to obtain the data required to create the PRBM of a spherical mechanism. The geometry and the material properties of the beam were defined as follows. Table 3.7. Object data for FEA of curved beams. Data Comments Arc angle, 15 30 45 60 75 and 90 Different arc angles were use in order to broaden the analysis of curved beams and develop a model that would work with any arc angle. Arc Length, s 10 The arc length was held constant throughout the analysis. Radius, R s The radius was defined as a function of the arc angle. Width w 20 s Height, h s 1 0 Modulus, E 169 GPa The ANSYS 3D beam element beam4 was used in the FEA model in order to study the bending axial and torsional deflections o ccurring in the beam. The geometry and boundary conditions of th e spherical model were specified as shown in figure 3.13: Node 1 was defined as the fixed center of th e sphere; thus, it was constrained in all directions.

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42 In order to have the opportunity to test the symmetry of the deflected beam, the arc was divided in four segment of equal length formed by nodes 2, 3, 4, 8, and 12. Node 4 was placed at the middle of the beam and nodes 8 and 12 were placed at the first and final quarters of the structure allowing collection of strategic data from these symmetric points on the arc. Node 2 de fined the fixed edge of the beam; therefore, it was constrained in all directions causing it to remain stationary th roughout the deflection motion; in addition, nodes 4, 8 and 12 were free allowing for the st udy and observation of the beamÂ’s structure as it was deflected. Moreover, node 3 was a guided end of the curved beam, where a horizontal buckling displacement load was applied and motion was prevented in the z -direction as were rotations about the y and x -axes. These boundary conditions allowed for translati on and rotation of the other no des, and for reaction forces and moments to be obtained at the constraine d nodes. Three orthogonal axes, a rotational reference frame were placed on nodes 4, 8, and 12 in order to track the motion of these nodes and to determine the twist about the be amÂ’s neutral axis. The reference frame at node 8 was defined by nodes 9, 10, and 11; the frame at node 4 was defined by nodes 5, 6, and 7; and the frame at node 12 was defined by nodes 13, 14, and 15.

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43 Figure 3.13. Shows the geometry, the load ing, and the nodes of the undeflected curved beam. Figure 3.14. The deflected curved beam.

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44 Once the beam was deflected the info rmation containing the nodesÂ’ reaction forces, rotations, and displacements on the x y and z directions was collected over a range of 70 load steps in an ANSYS data out put file. In addition, an analysis program was written; which used the output for the development and application of the new parametric beam model for curved beams th at will be introduced in section 3.5. This model produces a prediction of the kinematics and the elasticity of the deflected curved beam. 3.5 Curved Beam Kinematic Analysis In order to accurately us e the spherical PRBM to perform the kinematic and elastic analysis of a compliant curved beam with horizontal buckling loads, a specific analysis criterion was defined and refere nce frames were established based on the nomenclature developed by Saur abh Jagirdar [13]. The analys is criterion, the position, and the coordinate frames are related as follows: It was established that the beam deflects symmetrically meaning that the half and quarter segments of the beam are their mirro r images of each other. As a result we chose to analyze the segment from node 3 to node 12, which is a fourth of the beam and can be interpreted as a cant ilever curved beam. Then, the remaining beam pointsÂ’ characteristics coul d be calculated using symmetry. Moments about the x and y axes are equal and oppos ite at node 2 and node 3. The radial displacement of all nodes stays the same, meaning th at the radius of curvature does not chance as the beam deflects.

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45 Moments about the z axis at node 12 are close to zero meaning that there is no twist on the beam. The compliant curved beam, PG Figure 3.15 is described by using the following coordinate frames given in Table 3.8: Figure 3.15. The reference frames that de scribe the motion and orientation of positions on a compliant curved beam. The center of the sphere is defined by the O frame; the frames A B C and D are located on the surface of the sphere. The curved beam is denoted by the points P Q and G where P is on node 3; the free end of the beam, G is on node 2; the fixed end of the beam and Q is on node 12 which is the first quarter segment of the beam. The description of the coordinate frames is described on Table 3.6.

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46 Table 3.8. Coordinate frames position and orientation. Frame Frame Description O This is a non-moving frame at the center of the sphere and is established by the x y and z coordinate system program in the ASNSYS batch code. The O2 axis passes through the fix end of the beam at node 2 denote by G The O3 axis is normal to the plane containing the undeflected beam. The O1 axis lies on the x axis of the coordinate system. A This frame has identical orientation as O frame; however it is located at the fixed end of the beam on G is on the O2 axis. B This frame is in the same plane as O and A and underneath point Q in order to locate its deflected position in the A2A1 plane. C This frame is located on point Q and describes the movement of this point in the B3-B2 plane. D This frame is also located on point Q ; however, it is represented by the skew axis placed on node 12. This frame can be used to track the rotation about the C2 axis Table 3.9 provides a summary of the nomencl ature used in the analysis of the curved beam to facilitate th e understanding of the geometry.

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47 Table 3.9. Nomenclature. Variable Description Represents the translation of th e quarter segment, node 12, in the A2A1 plane, this transla tion is analogous to, a the translation in the x direction, in the planar model. Using symmetry is defined in equation (3.35). See Figure 3.18 tota l Represents the translation of th e half segment, node 4, in the A2A1 plane. total is defined in equation (3.36). Represents the translation of th e quarter segment, node 12, in the B3B2 plane, this transla tion is analogous to, b the translation in the y direction, in the planar model. Using symmetry is defined in equation (3.37). See Figure 3.16 tota l Represents the translation of th e half segment, node 4, in the B3B2 plane. totalis defined in equation (3.38). See Figure 3.18 Represents the angles that the nodes make with the x-axis as the beam deflects. See Figure 3.16 and 3.17 Represents the arc angle of the qua rter segment and it is defined on equation (3.39). tota l Represents the arc angle of the enti re curved beam and it is defined on equation (3.40). See Figure 3.14 Represent the total change of transl ation of the quarter segment, node 12, in the A2A1 plane and it is defined on equation (3.41). Represent the total change of tr anslation of the beam, in the A2A1 plane and it is defined on equation (3.42). Represents the amount of rotation th at the rigid model must undergo to match the deflection of the compliant curved beam. 0 Represents the Â“deflection of the b eam end about an axis normal to the tangent plane to the sphere at the beam endÂ” [12]. See Figure 3.15 s Represents the arc length of the curved beam. R Represents the radius of the sphere. 23 4 12 (3.36) 3 4 total (3.37)

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48 2 12 2 12 12 1 2 4 2 4 4 1 12tan 2 tan y x z y x z (3.38) 2 4 2 4 4 1tan y x ztotal (3.39) R s 412 (3.40) 124 R stotal (3.41) 12 12 12 12 12 f i (3.42) 4 2 23 . f total i total (3.43) Figures 3.16, 3.17, and 3.18 provide a gra phical explanation of the approach taken to define the geometry of the curved beam.

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49 Figure 3.16. The position coordinate s and angles of nodes 3 and 12. Figure 3.17. The position coordinate s and angles of nodes 4 and 8.

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50 Figure 3.18. The translation of th e quarter segment on the planes A2-A1 and B3-B2. The reference frames are defined by the matrices A B C, and D where each of their columns represents a basis vector The following equations represent the transformations that relate the frames. [ A ] is the identity matrix, 1 0 0 0 1 0 0 0 1 A (3.44) [ B ] is defined by A R BB (3.45) Where [ RB] (3.44) is a rotation matrix that transforms the vectors in plane A2-A1 such that they align with those in frame B

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51 1 0 0 0 2 cos 2 sin 0 2 sin 2 cos12 12 12 12 BR (3.46) [ C ] is given by B R CC (3.47) where [ RC] (3.46) is the rotation matrix that rotates vectors in the B3-B2 such that they align with the vectors in frame C 12 12 12 12cos sin 0 sin cos 0 0 0 1 CR (3.48) Equation (3.49) was used in order to calculate the amount of rotation that frame C must undergo about the C2 to match frame D and compute the value of0 1 C D RD (3.49) Where [ D ] is given by the coordinates of the frame placed at node 12 and [ RD] is specified in equation (3.50). 0 is found using the trace of [ RD] in equation (3.50). 0 0 0 0cos 0 sin 0 1 0 sin 0 cos DR (3.50) 2 1 cos1 0 DR Trace (3.51) Thus, the behavior of the curved compliant beam can be described by the parameters , , and 0 Using NapierÂ’s Rules for spherica l right triangles and trigonometric

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52 identities (See Figure 3.19) we can find relationships to define (3.52) and (3.53) in terms of , and Figure 3.19. The spherical right triangle formed by a curved beam. 12 12 12 12 1 12 *sin cos cos cos 1 tan 1 (3.52) 12 12 12 1 *sin tan tan (3.53) Once and are defined, one can apply th e statistical fitting technique explained in section 3.2 to represent the char acteristic radius fact or as a function of the moment load. Then, using the fit an d NapierÂ’s Rules the parameters , and can be found as follows.

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53 2 2 1 0 1 2 21 Âˆ Âˆ M b M b b M b M b M (3.54) 12 12 1 12Âˆ tan cos Âˆ cot cos 1 tan (3.55) 12 12 12 (3.56) 12 12 12 1 *Âˆ sin tan tan (3.57) 12 1 12sin Âˆ sin sin (3.58) Finally using symmetry one ca n calculate the parameters of the half segment node 4, , and its final output coordinate Z4, which represent the highest point reached by the mechanism for a given input. 12 42 (3.59) 12 42 (3.60) 12 42 (3.61) 12 12 12 12 12 42 sin 2 2 sin 2 2 cos R R R Z (3.62) 3.6 Kinematic Analysis Results La rge Deflection Curved Beams This section provides a summ ary of the kinematic analys is of the deflection of a curved beam under specific hor izontal buckling end-loads. The results shown in this section correspond to a curved beam with an arc angle, of 105 The reason why this angle was used is because it was the largest arc angle in the analysis; therefore, it was perceived as the worse can scen ario yielding to the highest errors. In other words, we based the model on this case because if the mo del works for a beam with an arc angle of

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54 105 it would work for any beam with a smaller arc angle producing smaller errors as the size of the arc angle decreased. The re sults of curved beams with angles 15 30 45 60 75 and 90 are shown on the results summary tables in the Appendix G. For a curved beam loaded with specif ic horizontal buckli ng loads a constant does not produce an accurate PRBM for the initial part of the beam deflection. Interestingly, this model behaves as Case 2 in the straight beams study; the constant PRBM approximation improves when the beam defl ects significantly, so that the force is perpendicular to the neutral axis of the beam. In the other hand, when is represented as a function of the load, as M Âˆ Âˆ with coefficient constants bi Table 3.10, the PRBM produces a more accurate prediction throughout the complete range of the deflection, meaning that regardless of the deflection magnitude or the dire ction in which the force is applied, this parametric model can provide an accurate prediction of the behavior of the beam Figure 3.20. Table 3.10. Values of the rational function coefficients for a curved beam with a horizontal buckling end-load. Rational function Coefficients b-2 -142.1823 b-1 78.0177 b0 -15.7772 b1 115.4554 b2 -34.7151

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55 Figure 3.20. versus moment for a curved beam with a horizontal buckling end-load. Figure 3.21 shows the parametric approxima tions of the end-beam deflection path of the curved beam loaded with specific bucklin g loads. Just as with the horizontal loaded straight beam, the new model is capable to improve the accuracy of the PRBM in comparison with a constant model.

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56 3.21. A horizontal end-load cu rved beam planar rotation versus out-of-plane rotation, Figure 3.22 shows the approximation of the deflected member PRBM angle, As with the straight beam case the new pa rametric model yields more accurate large deflection kinematic model prediction s than a PRBM using the constant

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57 Figure 3.22. Moment versus for a horizontal buckling end-load curved beam. A statistical analysis was performed on the curved beam PRBM as well; a twosided t test with a 95% confidence interval was developed; Fi gure 3.23 shows the characteristic radius factor within the 95% confidence interval. The coefficient of determination 2 BR the rational function prediction ma ximum error and total error, and the constant prediction total error in the deflecti on approximation were also calculated as shown in Table 3.11.

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58 Figure 3.23. The characteristic radius factor within the 95% confidence interval for a horizontal buckling end-load curved beam. Table 3.11. Results of statistic analysis fo r a horizontal buckling e nd-load curved beam. Rational function Coefficients 95% Confidence Interval b-2 -142.1823 4.1447 b-1 78.0177 3.0046 b0 -15.7772 1.3276 b1 115.4554 3.7958 b2 -34.7151 2.1215 Coefficient of Determination, 2 BR 99.44% Rational Function Prediction Maximum Error 1.1067e -3 % Rational Function Prediction Total Error 1.5795e-5% Constant Total Error 0.3843% In order to calculate the angle coefficient, C a plot of versus 0 was performed Figure 3.24; then, a lin ear fit was made to the curved yielding to the constant

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59 value of C needed to obtained the value of 0 in terms of the Pseudo-Rigid-Body angle, 3427 1 C (3.63) Figure 3.24. The approximation to 0 using the angle coefficient, C and the Pseudo-Rigid-Body angle, A PRBM for a curved beam under specific horizontal buckling end-loads has not been published before because PRBMs using a constant do not yield accurate predictions for this type of beams and loadi ng configurations. This new approach appears to fill the gap.

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60 Chapter 4 Elasticity of a PRBM for Curved Beams This Chapter uses the principle of virtual work used in to develop elasticity parameters such as the torque coefficient, T, and its components functions, Tf, and, Tm. According to Paul [25] Â“The ne t virtual work for all active forc es is zero if and only if an ideal mechanical system is in equilibrium. Â” The compliant mechanism analyzed in this thesis is assume to be ideal, meaning that the constrains on the mechanism do not do work. 0 W (4.1) 4.1 Principle of Virtual Work In order to apply the princi ple of virtual work to this model, first an arbitrary virtual linear displacement, z and an arbitrary virtua l angular displacement, must be defined as functions of the generalized coordinates (4.2). From (3.62) we expressed the virtual linear displacement as: k j R i R z Âˆ 0 Âˆ 2 cos Âˆ 2 sin3 (4.2) Then, the virtual work, FW due to the applied force, F and a virtual linear displacement, 3z is:

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61 3z F WF (4.3) In a like manner, the virtual work due to the applied moment, M and the virtual angular displacement, is: M WM (4.4) According to Howell [3] the PRBM can be used to model the compliant beamÂ’s resistance to bending by usi ng the stiffness coefficient, K, which represents a nondimensionalized torsional spring. This consta nt combined with the material properties and the geometry of the beam can be employe d to calculate the value of the PRBM spring constant, K (4.5). In order to calculated the value of the value of T the principle of virtual work and the PRBM concepts are used to establish force-deflection relationships for compliant mechanisms as descr ibe by Howell and Midha in [26]. R EI K K (4.5) Where R EI represents the non-di mensionalization factor. Moreover, all the store energy in the spri ngs of the PRBM must be taken into consideration in order to have a complete energy balance equation. Due to the symmetry of the deflected beam, the PRBM behaves as if it had four identical springs acting on each of its quarter segmentsÂ’ characteristic pi vots; therefore, the total energy store on the springs, sU is: d d K Us *4 (4.6)

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62 Where, K represents the spring constant and d d* represent the change of the PseudoRigid-Body Angle, as a function of the beamsÂ’ rotation in the plane A1A3, Because the quarter model is the obj ect being analyzed one can express d d* and sU in terms of the 12 12 *4 1d d d d (4.7) Therefore, 12 *d d K Us (4.8) Then, the total virtual work is 0 s M FU W W W (4.9) or 012 * 3 d d K M z F W (4.10) Solving for K and the resultant torque, T due to the force and the moment we obtained: 12 3 d d M z F K (4.11) 12 3 d d M z F K T (4.12) Furthermore, 12 d d can be found using equations (3.55) and NapierÂ’s Rules for spherical right triangles as follows:

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63 2 12 12 12 * 12 12 12 2 12Âˆ tan cos Âˆ cot Âˆ tan sin cos 1 Âˆ tan cos Âˆ cot sin sec 1 d d (4.13) Consequently, by substituting equation (4.5) in to equation (4.11) one can find a torque coefficient function, T, in terms of the virtual work. EI R d d M EI R d d z F T12 12 12 12 3 (4.14) Additionally, T can be separated into its component functions, fT (4.15) (the torque contributed by the force F ) and mT ( 4.16) (the torque c ontributed by the moment M ; then, the polynomial function of that best fits the torque is found as shown in Table 4.1 as the fits are shown in Figure 4.1. EI R d d z F Tf 12 12 3 (4.15) EI R d d M Tm 12 12 (4.16) m fT T T (4.17)

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64 Figure 4.1. The torque coefficient functionsT ,fT, and mT versus Table 4.1. Torque function coefficients. Function Function constant coefficients T Âˆ (*) -0.3244 *2+8.9923 *-0.1640 fT Âˆ (*) 9.1455 *-1.4521 mT Âˆ (*) -2.4524 *2+3.5246 *-0.0498 Once these component functions have b een established, one can determine the force (4.18) and the moment (4.19) applied a node 12 (quarter mode l). After these loads have been found, one can determine the values of the actual moment (4.21) and the actual force (4.22) applied at node 3. R EI T Ff 12Âˆ (4.18) 2 12Âˆ R EI T Mm (4.19)

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65 d d M M* 12 3 (4.20) d d F F* 12 3 (4.21) 4.2 Model Validation In order to validate the stiffness model a test was performed on a beam with different material properties a nd different cross-se ctional area from the original beam used to derived the model Table 4.2. Subsequently, the kinematic parameter was determined using the following expression 12 12 12 1 *Âˆ sin tan tan Where was the input rotation of the beam and was the kinematic model Âˆ. After that, using the force and moments loads were ca lculated using the torque function coefficients, fT, mT .in the following manner: First, the values of th e torque function confidents were calculated using as an input 0.0498 3.5246 + -2.4524 Âˆ 1.4521 + 9.1455 Âˆ 0.1640 8.9923 + -0.3244 Âˆ2 * * *2 m fT T T Then, using the non-dimensional torque coe fficient function and the dimensionalization factor R EI the force and the moment loads applied at node 12 were computed

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66 R EI T Ff 12Âˆ 2 12Âˆ R EI T Mm Finally, using the loads applied at no de 12 and the kinematic coefficient d d* the loads applied at node 3 were determined. d d M M* 12 3 d d F F* 12 3 In order to determine how well the loads were predicted by the model, the predicted loads were compared and plotted against the valu es of the loads acquired from the FEA analysis as shown in Figures 4.2 and 4.3. Finally, after doing an error analysis it was established that the elastic prediction erro r of the force and moment loads for a beam with an arc angle of 105 were 14.04 % and 14.53 % respectively, when compared to the data provided by the FEA analysis. However, when the error analysis was applied to the force and moment loads predicted fo r a beam with an arc angle of 15 it was found that the error decreased; th e error of the force and moment loads were 1.2218e-3% and 0.21 % respectively, when compared with the FEA data analysis. This suggests that the reason the errors on the predictions decrease for different arc angles, is because as the arc angles get smaller the spherical PRBM beha ves more as a planar PRBM simplifying the model and reducing the error.

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67 Table 4.2. Characteristics of the test beam. Arc angle, 15 30 45 60 75 90 105 Radius, R 100 m Arc length, s 10 Height, h s /15 Width, w w0.1 Modulus, E 180GPa 105 15 Maximum force load pred iction error 18.38% 1.8% Average force load prediction error 14.04 % 1.2218e-3% Maximum moment load prediction error 17.07% 1.57% Average moment load prediction error 14.53 % 0.21 % Figure 4.2. The loads predicted by the pa rametric model and the loads acquired from the FEA analysis for a beam with an arc length of 105

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68 Figure 4.3. The loads predicted by the pa rametric model and the loads acquired from the FEA analysis for a beam with an arc length of 15 4.3 Analysis of a Compliant Micro Helico -Kinematic Platform (MHKP) Device In order to analyze a compliant MHKP device with the new parametric beam model, the model was used to predict the mo tion and stiffness of a prototype with the properties given in Table 4.3, which w as designed and manufactured using the PolyMUMPs process [27] and is shown in Figures 4.4. Table 4.3. MHKP material properties a nd cross-sectional ar ea characteristics. Arc angle, 90 Radius, R 100 m Arc length, s s = R Height, h 2 m Width, w 2 m Modulus, E 169GPa

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69 Figure 4.4. Shows the MHKP device and outlines the curved beam being tested. After performing simulation we were able to determine the specific loads needed to actuate the device and th e output coordinates of the center of the beam, where z is the highest point reached by the beam once it is actuated as gi ven in Table 4.4. Table 4.4. MHKP device simulation results. Output coordinates x = 24.1031 y = 97.05217 z = 57.0909 (Vertical displacement) Finitial 143.35 N Ffinal 430.77 N Minitial 356.17 N m Mfinal 8.90 N m

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70 Chapter 5 Conclusion This chapter provides a conclusion to the wo rk developed in this thesis and gives a summary of the major contributions offered by this work. 5.1 Conclusion and Summary of Contributions This thesis provides a novel, more accur ate beam model for straight and curved compliant beams with vertical and horizont al buckling end-loads. The model uses a rational function to represent the characteristic radius factor as a function of the moment load, ÂˆM) which improves the accuracy and th e range of it when compared with previous models. The new parametric model is used to analyze the kinematics and elasticity of the complete deflection range of motion of both the straight and curved beams developing non-dimensional kinematic a nd elastic parameters such as the angle coefficient, C the characteristic radius factor, Âˆ the characteristic radius, Âˆl and the torque coefficients functions, T, Tm, Tf. In addition, the model is used to calculate the working loads on the curved beam using the input angle of rotation Furthermore, a compliant MHKP device was analyzed in order to determine specific buckling loads needed to actuate the device and the c oordinates of the center of

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71 its curved beams, so one c ould establish the highest point it could reac h once it was buckled. Finally, software codes were develope d in ANSYS and MATLAB in order to produce the new parametric model and pr ovide validation of its capabilities.

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72 References [1] Shigley, J. E. and Uicker, J. J. Theory of Machines and Mechanisms. 2nd. New York : McGraw-Hill, 1995. [2] Erdman, A.G. and Sandor, G.N. Mechanism Design: Analysis and Synthesis. 3rd. Upper Saddle River : Prentice Hall, 1997. Vol. 1. [3] Howell, Larry L. Compliant Mechanisms. New York : Wiley-Interscience, 2001. [4] Memsnet.org. MEMS and Nanotechnology Applications. [Online] [Cited: 1 12, 2009.] http://www.memsnet.org/mems/applications.html. [5] Clements, D. Implementing Compliant Mechanisms in Micro-Electro-Mechanical Systems (MEMS). Brigham Young University. Pr ovo, UT : s.n., 2000. M.S. Thesis. [6] A Micro Helico-kinematic Pla tform via Spherica l Crank-slider. Lusk, Craig P. and Howell, Larry L. 4, s.l. : ASME, 4 2008, Journal of Mechanical Design, Vol. 130. [7] Ortho-planar mechanisms. Pairse, J., Howell, L. and Magleby, S. 2000. Proceedings of the 2000 ASME Design E ngineering Technical Conferences, DETC2000/MECH-14193. pp. 1-15. [8] Lusk, Craig P. Ortho-Planar Mechanisms for Mi croelectromechanical Systems. Brigham Young University Provo, UT : s.n., 2005. Ph.D Dissertation. [9] A MEMS conical spring actuator array. Fukushinge, T., Hata, S. and Shimokohbe, A. April 2005, Journal of Microe lectromechanical Systems, Vol. 14, pp. 243. [10] MEMS Spatial Light Modulator Development at the Center of Adaptive Optics. Krulevitch, P., et al. 2003. SPIE. pp. 227-233. [11] Components, Building Blocks, and Demost rations of Spherical Mechanisms in Microelectromecanical Systems. Lusk, Craig P. and Howell, Larry L. s.l. : ASME, March 2008, Journal of Mech anical Design, Vol. 130. [12] Leon, Alejandro. A Pseudo-Rigid-Body Model fo r Spherical Mechanisms: The Kinematics and Elasticity of a Curved Compliant Beam. University of South Florida. Tampa, FL : s.n., 2007. M.S. Thesis.

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73 [13] Jagirdar, Saurabh. Kinematics of Curved Flexible Beam. University of South Florida. Tampa, FL : s.n., 2006. M.S. Thesis. [14] Parametric Deflection Approximations fo r End-Loaded, Large-Deflection Beams in Compliant Mechanisms. Howell, L. L. and Midha, A. 1, s.l. : ASME, March 1995, Journal of Mechanical Design, Vol. 117, pp. 156-165. [15] Preliminaries for a spherical complian t mechanism: Pseudo-Rigid-Body Model Kinematics. Lusk, C.P. and Jagirdar, S. Las Vegas, NV : s.n., 2007. ASME IDECT/CIE. [16] Spherical Kinematics in Contra st to Planar Kinematics. Chiang, C. H. Taipei, Taiwan : s.n., May 3, 1992, Mechanical Machine Theory, Vol. 27, pp. 243-250. [17] Spiegel, M. R. and Liu, J. Schaum's Outlines: Mathematical HandBook of Formulas and Tables. New York : McGraw-Hill, 1999. [18] Henderson, D. W. Esperiencing Geometry: In Euclidean, Spherical, and Hyperbolic spaces. 2nd edition. Upper Saddle Ri ver : Prentice Hall, 2001. [19] Newman, Donald J. Approximation with Rational Functions. Regional Conference series in Mathematics. Providence, Rhode Isla nd : Published for the Conference Board of Mathematical Sciences by the American Mathematical Society, 1979, 41, p. 52. [20] Contributors, Wikipedia. Pol ynomial and rational function modeling. Wikipedia, The Free Encyclopedia. [Online] 6 5, 2009. [Cited: January 5, 2009.] http://en.wikipedia.org/wiki/rat ional_function_modeling?oldid=217267798. [21] Mathbits.com. Statistics 2 Correlation Coefficient and Coefficient of Determination. [Online] [Cited: February 19, 2009.] http://mathbits.com/Mathbits/TISection/Statistics2/correlation.htm. [22] Montgomery, Douglas C, Runge r, George C and Faris, Norma. Engineering Statistics. New York : John Wiley & Sons, Inc, 2004. [23] Engineering Satatistics Handbook. NIST SAMTECH. [Online] [Cited: February 19, 2009.] : http://www.itl.nist.gov/di v898/handbook/eda/section3/eda352.htm [24] Methodology of compliant mechanisms and its current developments in applications: a review.(Report). Shuib, Solehuddin, Ridzwan and M.I.Z. Kadarman, Halim A. 3, 3 1, 2007, American Journal of Applied Sciences, Vol. 4, p. 160(8).

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74 [25] Paul, B. Kinematics and Dynamcis of Planar Machinery. Upper Saddle River : Prentice Hall, 1979. [26] The Development of Force-Deflection Re lationships for Compliant Mechanisms. Howell, Larry. L. and Midha, Ashok. s.l. : ASME, 1994. Machine Elements and Machine Dynamics. Vol. 71, pp. 501-508. [27] Koester, David, et al PolyMUMPs Design Hanbook. 2003.

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75 Appendices

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76 Appendix A: ANSYS Batch Code fo r a Vertical End-Loaded Beam !*********************************************************************** ********************** /CONFIG,NRES,10000 !/CWD,'C:\Documents and Sett ings\despinos\Desktop\Work' !*********************************************************************** ********************** !*********************************************************************** ********************** !******Set Up Model Variables**************************************************************** !*********************************************************************** ******************** !*DO,asp, .1,.7,.3 asp =.1 aspect = 10*asp !*DO,beamlenght,10,20,1 beamlenght=25 /PREP7 !LCLEAR, ALL !LDELE, ALL !KDELE, ALL R=25 PI=acos(-1.) h1=25 b1=100

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77 Appendix A: (continued) b2=10 h2=asp*b2 beamhigh=26.1-h2/2 !*********** Area properties ***************************************************************** A1 = h1*b1 Iz1= 1/12*b1*h1*h1*h1 Iy1= 1/12*h1*b1*b1*b1 E1= 300000 !*********************************************************************** ********************** A2= h2*b2 Iy2= 1/12*h2*b2*b2*b2 Iz2= 1/12*b2*h2*h2*h2 E2= 169000 !***********Declare an element type: Beam 4 (3D Elastic)************************************** ET,1,BEAM4 KEYOPT,1,2,1 KEYOPT,1,6,1 !***********Set Real Consta nts and Material Properties**************************************** R,1,A1,Iy1,Iz1,h1,b1, !Check on the assumptions being made R,2,A2,Iy2,Iz2,h2,b2, MPTEMP,1,0 MPDATA,EX,1,,E1 MPDATA,PRXY,1,,0.35 Material properties for material 1 and 2 MPTEMP,1,0 MPDATA,EX,2,,E2

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78 Appendix A: (continued) MPDATA,PRXY,2,,0.35 !*********************************************************************** ********************* !xcoor=R*cos((90-ar clength)*PI/180) !zcoor=R*sin((90-ar clength)*PI/180) zcoor=0 xcoor=beamlenght !**********Create Keypoints 1 throug 7: K( Point #, X-Coord, Y-Coord, ZCoord)**************** K,1, 0,0,0 K,2, beamlenght/4,0,0 K,3, beamlenght/2,0,0 K,4, 3*beamlenght/4,0,0 K,5, beamlenght,0,0 k,6, beamlenght/2, -1,h2/2 K,7, beamlenght/2, 0,h2/2+1 K,8, beamlenght/2, 1,h2/2 !*********Create Beam using Lines a nd an Arc and divide into segments************************ LSTR, 1,2 LSTR, 2,3 LSTR, 3,4 LSTR, 4,5 Draws lines connecting keypoints !1 through 6 LSTR, 3, 6 LSTR, 3, 7 LSTR, 3, 8 LESIZE, 5,,,1 LESIZE, 6,,,1 LESIZE, 7,,,1 LESIZE, 1,,,30 LESIZE, 2,,,30 LESIZE, 3,,,30 LESIZE, 4,,,30

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79 Appendix A: (continued) !***********MESH****************************************************** ********************** !rigid part, skew axis real, 1 Use real constant set 1 type, 1 Use element type 1 mat, 1 use material property set 1 LMESH, 5,7 mesh lines 3-5 !compliant part! real, 2 Use real constant set 2 type, 1 Use element type 1 mat, 2 use material property set 2 LMESH, 1,4 mesh line 1,3 !******Get Node Numb ers at chosen keypoints************************************************** ksel,s,kp,,1 nslk,s *get,nkp1,node,0,num,max nsel,all ksel,all ksel,s,kp,,2 nslk,s *get,nkp2,node,0,num,max !Retriev es a value and stores it as a scalar parameter or part of an array parameter nsel,all ksel,all ksel,s,kp,,3 nslk,s *get,nkp3,node,0,num,max nsel,all ksel,all ksel,s,kp,,4 nslk,s

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80 Appendix A: (continued) *get,nkp4,node,0,num,max !Retriev es a value and stores it as a scalar parameter or part of an array parameter 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 FINISH !*********************************************************************** ********************** !********************** SOLUTION ************************************************************* !*********************************************************************** ********************** /SOL ANTYPE,0 Sp ecifies the analysis type and restart status and "0" means that it Performs a static analysis. Valid for all degrees of freedom NLGEOM,1 Includes large-deflection effects in a static or full transient analysis !CNVTOL,U,,0.000001,,0 !CNVTOL,F,,0.0001,,0 !Sets convergence va lues for nonlinear analyses !*****************Constrains********************************************* ******* DK,1, ,0, , ,UX,UY,UZ,ROTX,ROTY,ROTz

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81 Appendix A: (continued) DK,5, ,0, , ,UY,ROTX,ROTz !*********************************************************************** ********* increments=250 loadsteps=250 DO,step,1,loadsteps,1 position1=(step-1)*be amhigh/increments !DDELE,12,ALL !*********************************************************************** ********* newz=h2/2+position1 dispz=newz+zcoor DK,5,UZ,dispz LSWRITE,step *ENDDO LSSOLVE,1,loadsteps FINISH !*********************************************************************** ********** !********GET RESULTS ************************************************************* !*********************************************************************** ********** !**************************Displacements nodes 2,3,5****************************** /POST1 !*DIM,rotx2,TABLE,loadSteps !*DIM,roty2,TABLE,loadSteps !*DIM,rotz2,TABLE,loadSteps

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83 Appendix A: (continued) *Do,nn,1,loadSteps set,nn !**************************Displacements node 2,3,5****************************** !*GET,rotx,Node,nkp2,ROT,X !*SET,rotx2(nn),rotx !*GET,roty,Node,nkp2,ROT,Y !*SET,roty2(nn),roty !*GET,rotz,Node,nkp2,ROT,Z !*SET,rotz2(nn),rotz !*GET,disX,Node,nkp2,U,X !*SET,disX2(nn),disX !*GET,disY,Node,nkp2,U,Y !*SET,disY2(nn),disY !*GET,disz,Node,nkp2,U,Z !*SET,disZ2(nn),disz !*GET,disX,Node,nkp3,U,X !*SET,disX3(nn),disX !*GET,disY,Node,nkp3,U,Y !*SET,disY3(nn),disY !*GET,disz,Node,nkp3,U,Z !*SET,disZ3(nn),disz *GET,roty,Node,nkp5,ROT,y *SET,roty5(nn),roty *GET,disX,Node,nkp5,U,X *SET,disX5(nn),disX *GET,disz,Node,nkp5,U,Z *SET,disZ5(nn),disz !**************************Reactions forces node 1******************************* *GET,momx,Node,nkp1,RF,MX

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84 Appendix A: (continued) *SET,momx1(nn),momx *GET,fx,Node,nkp1,RF,FX *SET,fx1(nn),fx *GET,momy,Node,nkp1,RF,MY *SET,momy1(nn),momy *GET,fy,Node,nkp1,RF,FY *SET,fy1(nn),fy *GET,momz,Node,nkp1,RF,MZ *SET,momz1(nn),momz *GET,fz,Node,nkp1,RF,FZ *SET,fz1(nn),fz !**************************Reactions forces node 2******************************* !*GET,momx,Node,nkp2,RF,MX !*SET,momx2(nn),momx !*GET,fx,Node,nkp2,RF,FX !*SET,fx2(nn),fx !*GET,momy,Node,nkp2,RF,MY !*SET,momy2(nn),momy !*GET,fy,Node,nkp2,RF,FY !*SET,fy2(nn),fy !*GET,momz,Node,nkp2,RF,MZ !*SET,momz2(nn),momz !*GET,fz,Node,nkp2,RF,FZ !*SET,fz2(nn),fz !**************************Reactions forces node 5******************************* *GET,momx,Node,nkp5,RF,MX *SET,momx5(nn),momx *GET,fx,Node,nkp5,RF,FX *SET,fx5(nn),fx *GET,momy,Node,nkp5,RF,MY *SET,momy5(nn),momy *GET,fy,Node,nkp5,RF,FY *SET,fy5(nn),fy *GET,momz,Node,nkp5,RF,MZ *SET,momz5(nn),momz *GET,fz,Node,nkp5,RF,FZ *SET,fz5(nn),fz

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85 Appendix A: (continued) *ENDDO /output,forcevertical2% _asp%aspect%,txt,,Append !*********************************************************************** *************** !***************FILE HEADER: BEAM DATA************************************************* !*********************************************************************** *************** *MSG,INFO,'h2','b2 ','R','E','beamle nght','beamhigh' %-8C %-8C %-8C %-8C %-8C %-8C *VWRITE,h2,b2,R,E2,beamlength,beamhigh %16.8G %-16.8G %-16.8G %-16.8G %-16.8G %-16.8G !*********************************************************************** ************ !**************DISPLACEMENT DATA SET************************************************ !*********************************************************************** ************ !*MSG,INFO,'rotX2','rotY2','ro tZ2','disX2','disY2','disZ2' !%-8C %-8C %-8C %-8C %-8C %-8C !*VWRITE,rotx2(1),roty2( 1),rotz2(1),disX2(1),disY2(1),disZ2(1) !%16.8G %-16.8G %-16.8G %16.8G %-16.8G %-16.8G *MSG,INFO,'roty5','disX5','disZ5' %-8C %-8C %-8C *VWRITE,roty5(1),disX5(1),disZ5(1) %16.8G %-16.8G %-16.8G !*********************************************************************** ************

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86 Appendix A: (continued) !**************REACTIONS AT NODE 2************************************************** !*********************************************************************** ************ !*MSG,INFO,'momx2','momy2','momz2','fx2','fy2','fz2' !%-8C %-8C %-8C %-8C %-8C %-8C !*VWRITE,momx2(1),momy2(1),mom z2(1),fx2(1),fy2(1),fz2(1) !%16.8G %-16.8G %-16.8G %16.8G %-16.8G %-16.8G !*********************************************************************** ************ !**************REACTIONS AT NODE 5************************************************** !*********************************************************************** ************ *MSG,INFO,'momx5','momy5', 'momz5','fx5', 'fy5','fz5' %-8C %-8C %-8C %-8C %-8C %-8C *VWRITE,momx5(1),momy5(1),mom z5(1),fx5(1),fy5(1),fz5(1) %16.8G %-16.8G %-16.8G %16.8G %-16.8G %-16.8G !*********************************************************************** ************ !**************REACTIONS AT NODE 1************************************************** !*********************************************************************** ************ *MSG,INFO,'momx1','momy1', 'momz1','fx1', 'fy1','fz1' %-8C %-8C %-8C %-8C %-8C %-8C *VWRITE,momx1(1),momy1(1),mom z1(1),fx1(1),fy1(1),fz1(1) %16.8G %-16.8G %-16.8G %16.8G %-16.8G %-16.8G !*********************************************************************** ************

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87 Appendix A: (continued) !*********************************************************************** ************ !*********************************************************************** ************ /output FINISH *ENDDO *ENDDO

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89 Appendix B: (continued) momy5 = DATA2(:,2); momz5 = DATA2(:,3); fx5 = DATA2(:,4); fy5 = DATA2(:,5); fz5 = DATA2(:,6); momx1 = DATA3(:,1); momy1 = DATA3(:,2); momz1 = DATA3(:,3); fx1 = DATA3(:,4); fy1 = DATA3(:,5); fz1 = DATA3(:,6); %%%%%%%DEFINING l,a,b,gamma,Captheta,theta0,torque%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% l = 25; a= l+disx5; b =disz5; gamma = -(a.^2 2*a*l+l^2+b.^2)./(2*a*l-2*l^2); theta = atan2(b,a-(1-gamma)*l); theta(1) =0; Torque = -momy1; theta0 = roty5; E = 169000; h = 1; b1 = 10; l = 25; I = (b1*h^3)/12; M = (momy1*l^2)/(E*I); % Nondimensionalization M_max = max(M); %Scales moment between 0 and 1 M = M/M_max; R = normrnd(0.1,.1,[size(M)]); X = [ones(size(M(2:end))) 1./M(2:end) 1./M(2:end).^2 1.*M(2:end) 1.*M(2:end).^2]; Y = gamma(2:end); B = inv(X'*X)*X'*Y; %Gamma = 1./(B(1)+B(2)./M+B(3)./M.^2+B(4)./M.^3+B(5)./M.^4+B(6)./M.^5); Gamma = B(1)+B(2)./M+B(3)./M.^2 +B(4).*M +B(5).*M.^2; Csgamma=polyfit(M*M_max,gamma,0); Theta = atan2(b,a-(1-Gamma)*l);

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90 Appendix B: (continued) ThetaC= atan2(b,a-(1-Csgamma)*l); epsilon = gamma(2:end)-Gamma(2:end); SST = sum((gamma(2:end)-mean(gamma(2:end))).^2); SSE = sum(epsilon.^2); s_squared = SSE/(length(gamma(2:end))-length(B)); s = sqrt(s_squared); rsqrd = 1SSE/SST; Var_b = inv(X'*X)*s_squared; X_ortho = X'*X; [V,D] = eig(X_ortho); X2(:,1) = X*V(:,1); X2(:,2) = X*V(:,2); X2(:,3) = X*V(:,3); X2(:,4) = X*V(:,4); X2(:,5) = X*V(:,5); C = inv(X2'*X2)*X2'*Y; %Gamma2 = C(1)*X2(:,1)+C(2)*X2(:,2)+C(3)*X2(:,3)+C(4)*X2(:,4)+C(5)*X2(:,5); Gamma2 = C(1)*X2(:,1)+C(2)*X2(:,2)+C(3)*X2(:,3)+C(4)*X2(:,4)+C(5)*X2(:,5); Theta2 = atan2(b(2:end),a(2:end)-(1-Gamma2)*l); epsilon2 = gamma(2:end)-Gamma2; SSE2 = sum(epsilon2.^2); s_squared2 = SSE2/(length(gamma(2:end))-length(C)); s2 = sqrt(s_squared2); rsqrd2 = 1SSE2/SST; Var_c = inv(X2'*X2)*s_squared2; B1_prime = C(1)*V(1,1)+C(2)*V(1,2)+C(3)*V(1,3)+C(4)*V(1,4)+C(5)*V(1,5); % compare with B(1) B2_prime = C(1)*V(2,1)+C(2)*V(2,2)+C(3)*V(2,3)+C(4)*V(2,4)+C(5)*V(2,5); % compare with B(2) B3_prime = C(1)*V(3,1)+C(2)*V(3,2)+C(3)*V(3,3)+C(4)*V(3,4)+C(5)*V(3,5); % compare with B(3) B4_prime = C(1)*V(4,1)+C(2)*V(4,2)+C(3)*V(4,3)+C(4)*V(4,4)+C(5)*V(4,5); % compare with B(4) B5_prime = C(1)*V(5,1)+C(2)*V(5,2)+C(3)*V(5,3)+C(4)*V(5,4)+C(5)*V(5,5); % compare with B(5)

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91 Appendix B: (continued) B_var(1) = abs(Var_c(1,1)*V(1,1)+Var_c(2,2)*V(1,2)+Var_c(3,3)*V(1,3)+Var_c(4,4)*V( 1,4)+Var_c(5,5)*V(1,5)); B_var(2) = abs(Var_c(1,1)*V(2,1)+Var_c(2,2)*V(2,2)+Var_c(3,3)*V(2,3)+Var_c(4,4)*V( 2,4)+Var_c(5,5)*V(2,5)); B_var(3) = abs(Var_c(1,1)*V(3,1)+Var_c(2,2)*V(3,2)+Var_c(3,3)*V(3,3)+Var_c(4,4)*V( 3,4)+Var_c(5,5)*V(3,5)); B_var(4) = abs(Var_c(1,1)*V(4,1)+Var_c(2,2)*V(4,2)+Var_c(3,3)*V(4,3)+Var_c(4,4)*V( 4,4)+Var_c(5,5)*V(4,5)); B_var(5) = abs(Var_c(1,1)*V(5,1)+Var_c(2,2)*V(5,2)+Var_c(3,3)*V(5,3)+Var_c(4,4)*V( 5,4)+Var_c(5,5)*V(5,5)); B_std = sqrt(B_var); % 95 % 2 sided confidence interval ie mean + or interval t_statistic = tinv(.975,length(Y)-length(B)); CI = t_statistic*B_std; Gamma_minus = B(1)-CI(1)+(B(2)-CI(2))./M+(B(3)-CI(3))./M.^2 +(B(4)CI(4)).*M +(B(5)-CI(5)).*M.^2; Gamma_plus = B(1)+CI(1)+(B(2)+CI(2))./M+(B(3)+CI(3))./M.^2 +(B(4)+CI(4)).*M +(B(5)+CI(5)).*M.^2; %%%%%%%%%%%FIGURE 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure(4) plot([Gamma(2:end) Gamma_minus(2:end) Gamma_plus(2:end)]); %%%%%%%%%%%FIGURE 1 a vs b %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure(1) clf plot(a/l,b/l, 'b*' ,((1Gamma2(2:end))+Gamma2(2:end).*cos(Theta2(2:end))),Gamma2(2:end).*sin(Th eta2(2:end)), 'R' ,((1Csgamma)+Csgamma*cos(Theta2)),Csgamma*sin(Theta2), 'G-' );

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92 Appendix B: (continued) %%%%%%%%%%%FIGURE 2 M vs theta %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure(2) clf plot(M,theta*180/pi, 'b*' ,M(2:end),Theta2*180/pi, 'R' ,M(2:end),ThetaC(2:e nd)*180/pi, 'G*-' ); %%%%%%%%%%%FIGURE 3 M vs gamma %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure (3) plot(gamma,M*M_max, 'b*' ,Gamma2,M(2:end)*M_max, 'R' ,Csgamma,M*M_max, 'G' ); mytexstr = '$\frac{M l^2}{EI}$' ; Gc= ylabel(mytexstr, 'interpreter' 'latex' 'fontsize' ,10, 'units' 'norm' ); G4c = legend( 'Data' '\gamma*' '\gamma' ); %%%%%%%%%%%ERROR %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% erroro = ( a/l ((1-Gamma)+Gamma.*cos(Theta))).^2 + (b/l Gamma.*sin(Theta)).^2; %disp_mag = ((((1-Gamma)+Gamma.*cos(Theta))a(round(end/3))/l).^2+(Gamma.*sin(Theta)-b(round(end/3))/l).^2).^.5; total_erroro = trapz(M(2:end),erroro(2:end)); %+trapz(Gamma(2:end),erroro(2:end)); %total_erroro = sqrt(max(erroro(2:end)./diff(M))) rsqrd Csgamma total_erroro CI

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93 Appendix C: ANSYS Batch Co de for a Specific Horizont al Buckling End-Loaded Beam !*********************************************************************** ********************** /CONFIG,NRES,10000 !/CWD,'C:\Documents and Sett ings\despinos\Desktop\Work' !*********************************************************************** ********************** !*********************************************************************** ********************** !******Set Up Model Variables**************************************************************** !*********************************************************************** ********************** !*DO,asp, .1,.7,.3 asp =.1 aspect = 10*asp !*DO,beamlenght,10,20,1 beamlenght=100 /PREP7 !LCLEAR, ALL !LDELE, ALL !KDELE, ALL R=100 PI=acos(-1.) h1=25 b1=100 b2=10

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94 Appendix C: (continued) h2=asp*b2 !*********** Area properties ***************************************************************** A1 = h1*b1 Iz1= 1/12*b1*h1*h1*h1 Iy1= 1/12*h1*b1*b1*b1 E1= 300000 !*********************************************************************** ********************** A2= h2*b2 Iy2= 1/12*h2*b2*b2*b2 Iz2= 1/12*b2*h2*h2*h2 E2= 169000 !***********Declare an element type: Beam 4 (3D Elastic)************************************** ET,1,BEAM4 KEYOPT,1,2,1 KEYOPT,1,6,1 !***********Set Real Consta nts and Material Properties**************************************** R,1,A1,Iy1,Iz1,h1,b1, !Check on the assumptions being made R,2,A2,Iy2,Iz2,h2,b2, MPTEMP,1,0 MPDATA,EX,1,,E1 MPDATA,PRXY,1,,0.35 Material properties for material 1 and 2 MPTEMP,1,0 MPDATA,EX,2,,E2 MPDATA,PRXY,2,,0.35

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95 Appendix C: (continued) ************************************************************************ ******************** xcoor=beamlenght !**********Create Keypoints 1 throug 7: K( Point #, X-Coord, Y-Coord, ZCoord)**************** K,1, 0,0,0 K,2, beamlenght/4,0,h2/4 K,3, beamlenght/2,0,h2/2 K,4, 3*beamlenght/4,0,h2/4 K,5, beamlenght,0,0 k,6, beamlenght/2, -1,h2/2 K,7, beamlenght/2, 0,h2/2+1 K,8, beamlenght/2, 1,h2/2 !*********Create Beam using Lines a nd an Arc and divide into segments************************ LSTR, 1,2 LSTR, 2,3 LSTR, 3,4 LSTR, 4,5 Draws lines connecting keypoints 1 through 6 LSTR, 3, 6 LSTR, 3, 7 LSTR, 3, 8 LESIZE, 5,,,1 LESIZE, 6,,,1 LESIZE, 7,,,1 LESIZE, 1,,,30 LESIZE, 2,,,30 LESIZE, 3,,,30 LESIZE, 4,,,30 !***********MESH****************************************************** **********************

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96 Appendix C: (continued) !rigid part, skew axis real, 1 Use real constant set 1 type, 1 Use element type 1 mat, 1 use material property set 1 LMESH, 5,7 mesh lines 3-5 !compliant part! real, 2 Use real constant set 2 type, 1 Use element type 1 mat, 2 use material property set 2 LMESH, 1,4 mesh line 1,3 !******Get Node Numb ers at chosen keypoints************************************************** ksel,s,kp,,1 nslk,s *get,nkp1,node,0,num,max nsel,all ksel,all ksel,s,kp,,2 nslk,s *get,nkp2,node,0,num,max !Retrieves a value and stores it as a scalar parameter or part of an array parameter nsel,all ksel,all ksel,s,kp,,3 nslk,s *get,nkp3,node,0,num,max nsel,all ksel,all ksel,s,kp,,4 nslk,s *get,nkp4,node,0,num,max !Retriev es a value and stores it as a scalar parameter or part of an array parameter

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97 Appendix C: (continued) 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 FINISH !*********************************************************************** ********************** !********************** SOLUTION ************************************************************* !*********************************************************************** ********************** /SOL ANTYPE,0 Sp ecifies the analysis type and restart status and "0" means that it Performs a static analysis. Valid for all degrees of freedom NLGEOM,1 In cludes large-deflection effects in a static or full transient analysis !CNVTOL,U,,0.000001,,0 !CNVTOL,F,,0.0001,,0 !Sets convergence va lues for nonlinear analyses !*****************Constrains********************************************* ******* DK,1, ,0, , ,UX,UY,UZ,ROTX,ROTY,ROTz

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98 Appendix C: (continued) DK,5, ,0, , ,UZ, UY,ROTY,ROTX,ROTz !*********************************************************************** ********* increments=3000 loadsteps=300 *DO,step,1,loadsteps,1 position=(step-1)*beamlenght/increments !DDELE,12,ALL !*********************************************************************** ********* newx=beamlenght-position dispx=newx-xcoor DK,5,UX,dispx LSWRITE,step *ENDDO increments=400 loadsteps=284 *DO,step,41,loadsteps,1 position=(step-1)*beamlenght/increments !DDELE,12,ALL !*********************************************************************** ********* newx=beamlenght-position dispx=newx-xcoor DK,5,UX,dispx LSWRITE,step+260 *ENDDO LSSOLVE,1,loadsteps+260

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101 Appendix C: (continued) *GET,disY,Node,nkp5,U,Y *SET,disY5(nn),disY *GET,disz,Node,nkp5,U,Z *SET,disZ5(nn),disz !**************************Reactions forces node 1******************************* *GET,momx,Node,nkp1,RF,MX *SET,momx1(nn),momx *GET,fx,Node,nkp1,RF,FX *SET,fx1(nn),fx *GET,momy,Node,nkp1,RF,MY *SET,momy1(nn),momy *GET,fy,Node,nkp1,RF,FY *SET,fy1(nn),fy *GET,momz,Node,nkp1,RF,MZ *SET,momz1(nn),momz *GET,fz,Node,nkp1,RF,FZ *SET,fz1(nn),fz !**************************Reactions forces node 2******************************* *GET,momx,Node,nkp2,RF,MX *SET,momx2(nn),momx *GET,fx,Node,nkp2,RF,FX *SET,fx2(nn),fx *GET,momy,Node,nkp2,RF,MY *SET,momy2(nn),momy *GET,fy,Node,nkp2,RF,FY *SET,fy2(nn),fy *GET,momz,Node,nkp2,RF,MZ *SET,momz2(nn),momz *GET,fz,Node,nkp2,RF,FZ *SET,fz2(nn),fz !**************************Reactions forces node 5******************************* *GET,momx,Node,nkp5,RF,MX *SET,momx5(nn),momx *GET,fx,Node,nkp5,RF,FX *SET,fx5(nn),fx

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102 Appendix C: (continued) *GET,momy,Node,nkp5,RF,MY *SET,momy5(nn),momy *GET,fy,Node,nkp5,RF,FY *SET,fy5(nn),fy *GET,momz,Node,nkp5,RF,MZ *SET,momz5(nn),momz *GET,fz,Node,nkp5,RF,FZ *SET,fz5(nn),fz *ENDDO /output,forceplana2%_as p%aspect%,txt,,Append !*********************************************************************** *************** !***************FILE HEADER: BEAM DATA************************************************* !*********************************************************************** *************** *MSG,INFO,'h2','b2', 'R','E','beamlenght' %-8C %-8C %-8C %-8C %-8C *VWRITE,h2,b2,R,E2,beamlength %16.8G %-16.8G %-16.8G %-16.8G %-16.8G !*********************************************************************** ************ !**************DISPLACEMENT DATA SET************************************************ !*********************************************************************** ************ *MSG,INFO,'rotX2','rotY2','rotZ2','disX2','disY2','disZ2' %-8C %-8C %-8C %-8C %-8C %-8C *VWRITE,rotx2(1),roty2(1),rotz2( 1),disX2(1),disY2(1),disZ2(1) %16.8G %-16.8G %-16.8G %16.8G %-16.8G %-16.8G *MSG,INFO,'disX3','disY3','disZ3','disX5','disY5','disZ5' %-8C %-8C %-8C %-8C %-8C %-8C

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103 Appendix C: (continued) *VWRITE,disX3(1),disY3(1),disZ3( 1),disX5(1),disY5(1),disZ5(1) %16.8G %-16.8G %-16.8G %16.8G %-16.8G %-16.8G !*********************************************************************** ************ !**************REACTIONS AT NODE 2************************************************** !*********************************************************************** ************ *MSG,INFO,'momx2','momy2', 'momz2','fx2', 'fy2','fz2' %-8C %-8C %-8C %-8C %-8C %-8C *VWRITE,momx2(1),momy2(1),mom z2(1),fx2(1),fy2(1),fz2(1) %16.8G %-16.8G %-16.8G %16.8G %-16.8G %-16.8G !*********************************************************************** ************ !**************REACTIONS AT NODE 5************************************************** !*********************************************************************** ************ *MSG,INFO,'momx5','momy5', 'momz5','fx5', 'fy5','fz5' %-8C %-8C %-8C %-8C %-8C %-8C *VWRITE,momx5(1),momy5(1),mom z5(1),fx5(1),fy5(1),fz5(1) %16.8G %-16.8G %-16.8G %16.8G %-16.8G %-16.8G !*********************************************************************** ************ !**************REACTIONS AT NODE 1************************************************** !*********************************************************************** ************ *MSG,INFO,'momx1','momy1', 'momz1','fx1', 'fy1','fz1' %-8C %-8C %-8C %-8C %-8C %-8C

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104 Appendix C: (continued) *VWRITE,momx1(1),momy1(1),mom z1(1),fx1(1),fy1(1),fz1(1) %16.8G %-16.8G %-16.8G %16.8G %-16.8G %-16.8G !*********************************************************************** ************ /output FINISH *ENDDO *ENDDO

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PAGE 117

106 Appendix D: (continued) %%%%%%%%%%%%%%%%%%%%%%%%%%FORCES, DISPLACEMENTS AND MOMENTD%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%% roty2 = DATA1(:,2); disx2 = DATA1(:,4); disz2 = DATA1(:,6); disx3 = DATA2(:,1); disz3 = DATA2(:,3); disx5 = DATA2(:,4); momx2 = DATA3(:,1); momy2 = DATA3(:,2); momz2 = DATA3(:,3); fx2 = DATA3(:,4); fy2 = DATA3(:,5); fz2 = DATA3(:,6); momx5 = DATA4(:,1); momy5 = DATA4(:,2); momz5 = DATA4(:,3); fx5 = DATA4(:,4); fy5 = DATA4(:,5); fz5 = DATA4(:,6); momx1 = DATA5(:,1); momy1 = DATA5(:,2); momz1 = DATA5(:,3); fx1 = DATA5(:,4); fy1 = DATA5(:,5); fz1 = DATA5(:,6); %%%%%%%DEFINING l,a,b,gamma,Captheta,theta0,torque%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% l = 25; a= l+disx2; b =disz2; gamma = -(a.^2 2*a*l+l^2+b.^2)./(2*a*l-2*l^2); theta = atan2(b,a-(1-gamma)*l); theta(1) =0; Torque = -momy1; theta0 = roty2; E = 169000; h = 1; b1 = 10; l = 25; I = (b1*h^3)/12; M = (momy1*l^2)/(E*I); M_max = max(M); M = M/M_max; R = normrnd(0.1,.1,[size(M)]);

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107 Appendix D: (continued) X = [ones(size(M(2:end))) 1./M(2:end) 1./M(2:end).^2 1.*M(2:end) 1.*M(2:end).^2]; Y = gamma(2:end); B = inv(X'*X)*X'*Y; %Gamma = 1./(B(1)+B(2)./M+B(3)./M.^2+B(4)./M.^3+B(5)./M.^4+B(6)./M.^5); Gamma = B(1)+B(2)./M+B(3)./M.^2 +B(4).*M +B(5).*M.^2; Csgamma=polyfit(M(2:end)*M_max,gamma(2:end),0); %%%%%%%%% Theta = atan2(b,a-(1-Gamma)*l); ThetaC= atan2(b,a-(1-Csgamma)*l); figure (5) clf plot(M*M_max,gamma) epsilon = gamma(2:end)-Gamma(2:end); SST = sum((gamma(2:end)-mean(gamma(2:end))).^2); SSE = sum(epsilon.^2); s_squared = SSE/(length(gamma(2:end))-length(B)); s = sqrt(s_squared); rsqrd = 1SSE/SST; Var_b = inv(X'*X)*s_squared; X_ortho = X'*X; [V,D] = eig(X_ortho); X2(:,1) = X*V(:,1); X2(:,2) = X*V(:,2); X2(:,3) = X*V(:,3); X2(:,4) = X*V(:,4); X2(:,5) = X*V(:,5); C = inv(X2'*X2)*X2'*Y; %Gamma2 = C(1)*X2(:,1)+C(2)*X2(:,2)+C(3)*X2(:,3)+C(4)*X2(:,4)+C(5)*X2(:,5); Gamma2 = C(1)*X2(:,1)+C(2)*X2(:,2)+C(3)*X2(:,3)+C(4)*X2(:,4)+C(5)*X2(:,5); Theta2 = atan2(b(2:end),a(2:end)-(1-Gamma2)*l); epsilon2 = gamma(2:end)-Gamma2; SSE2 = sum(epsilon2.^2); s_squared2 = SSE2/(length(gamma(2:end))-length(C)); s2 = sqrt(s_squared2);

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108 Appendix D: (continued) rsqrd2 = 1SSE2/SST; Var_c = inv(X2'*X2)*s_squared2; sc(1) = sqrt(Var_c(1,1)); sc(2) = sqrt(Var_c(2,2)); sc(3) = sqrt(Var_c(3,3)); sc(4) = sqrt(Var_c(4,4)); sc(5) = sqrt(Var_c(5,5)); B1_prime = C(1)*V(1,1)+C(2)*V(1,2)+C(3)*V(1,3)+C(4)*V(1,4)+C(5)*V(1,5); % compare with B(1) B2_prime = C(1)*V(2,1)+C(2)*V(2,2)+C(3)*V(2,3)+C(4)*V(2,4)+C(5)*V(2,5); % compare with B(2) B3_prime = C(1)*V(3,1)+C(2)*V(3,2)+C(3)*V(3,3)+C(4)*V(3,4)+C(5)*V(3,5); % compare with B(3) B4_prime = C(1)*V(4,1)+C(2)*V(4,2)+C(3)*V(4,3)+C(4)*V(4,4)+C(5)*V(4,5); % compare with B(4) B5_prime = C(1)*V(5,1)+C(2)*V(5,2)+C(3)*V(5,3)+C(4)*V(5,4)+C(5)*V(5,5); % compare with B(5) B_var(1) = sum(sc.*V(1,:)).^2; B_var(2) = sum(sc.*V(2,:)).^2; B_var(3) = sum(sc.*V(3,:)).^2; B_var(4) = sum(sc.*V(4,:)).^2; B_var(5) = sum(sc.*V(5,:)).^2; B_std = sqrt(B_var); % 95 % 2 sided confidence interval ie mean + or interval t_statistic = tinv(.975,length(Y)-length(B)); CI = t_statistic*B_std Gamma_minus = B(1)-CI(1)+(B(2)-CI(2))./M+(B(3)-CI(3))./M.^2 +(B(4)CI(4)).*M +(B(5)-CI(5)).*M.^2; Gamma_plus = B(1)+CI(1)+(B(2)+CI(2))./M+(B(3)+CI(3))./M.^2 +(B(4)+CI(4)).*M +(B(5)+CI(5)).*M.^2; %%%%%%%%%%%FIGURE 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure(4) plot([Gamma(2:end) Gamma_minus(2:end) Gamma_plus(2:end)]);

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109 Appendix D: (continued) %%%%%%%%%%%FIGURE 1 a vs b %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure(1) clf plot(a/l,b/l, 'b*' ,((1Gamma2(2:end))+Gamma2(2:end).*cos(Theta2(2:end))),Gamma2(2:end).*sin(Th eta2(2:end)), 'R' ,((1Csgamma)+Csgamma*cos(Theta2)),Csgamma*sin(Theta2), 'G-' ); %%%%%%%%%%%FIGURE 2 M vs theta %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure(2) clf plot(M,theta*180/pi, 'b*' ,M(2:end),Theta2*180/pi, 'R' ,M(2:end),ThetaC(2:e nd)*180/pi, 'G-' ); %%%%%%%%%%%FIGURE 3 M vs gamma %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure (3) plot(gamma,M*M_max, 'b*' ,Gamma2,M(2:end)*M_max, 'R' ,Csgamma,M*M_max, 'Gs' ); mytexstr = '$\frac{M l^2}{EI}$' ; Gc= ylabel(mytexstr, 'interpreter' 'latex' 'fontsize' ,10, 'units' 'norm' ); G4c = legend( 'Data' '\gamma*' '\gamma' ); %%%%%%%%%%%ERROR %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% erroro = ( a(2:end)/l ((1-Gamma2)+Gamma2.*cos(Theta2))).^2 + (b(2:end)/l Gamma2.*sin(Theta2)).^2; %disp_mag = ((((1-Gamma2)+Gamma2.*cos(Theta2))a(round(end/3))/l).^2+(Gamma2.*sin(Theta2)-b(round(end/3))/l).^2).^.5; total_erroro = trapz(M(2:end),erroro); %+trapz(Gamma2(2:end),erroro(2:end)); %total_erroro = sqrt(max(erroro(2:end)./diff(M))) rsqrd2 Csgamma total_erroro

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110 Appendix E: ANSYS Batch Code for a Speci fic Horizontal Buckling End-Loaded Curved Beam !*********************************************************************** ********************** /CONFIG,NRES,10000 !/CWD,'C:\Documents and Sett ings\despinos\Desktop\Work' !/INPUT,'C:\Documents and Settings\despinos\Desktop\ThesisDiego\ AnsysCode\curvebeam3diffarcs','txt' !*********************************************************************** ********************** !*********************************************************************** ********************** !******Set Up Model Variables**************************************************************** !*********************************************************************** ********************** !*DO,asp, .1,.7,.3 asp =.1 aspect = 10*asp !*DO,beamlenght,10,20,1 *DO,LAMBDAdg,15,105,15 R=100 Lambda=LAMBDAdg*PI/180 arclength=R*Lambda /PREP7 LCLEAR, ALL LDELE, ALL KDELE, ALL PI=acos(-1.) h1=.1 b1=1 b2=5.7 h2=2

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111 Appendix E: (continued) !*********** Area properties ***************************************************************** A1 = h1*b1 Iz1= 1/12*b1*h1*h1*h1 Iy1= 1/12*h1*b1*b1*b1 E1= 300000 !*********************************************************************** ********************** A2= h2*b2 Iy2= 1/12*h2*b2*b2*b2 Iz2= 1/12*b2*h2*h2*h2 E2= 169000 !***********Declare an element type: Beam 4 (3D Elastic)************************************** ET,1,BEAM4 KEYOPT,1,2,1 KEYOPT,1,6,1 !***********Set Real Consta nts and Material Properties**************************************** R,1,A1,Iy1,Iz1,h1,b1, !Check on the assumptions being made R,2,A2,Iy2,Iz2,h2,b2, MPTEMP,1,0 MPDATA,EX,1,,E1 MPDATA,PRXY,1,,0.35 Material properties for material 1 and 2 MPTEMP,1,0 MPDATA,EX,2,,E2

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112 Appendix E: (continued) MPDATA,PRXY,2,,0.35 !*********************************************************************** ********************* xcoor=R*cos((90*PI/180)-(arclength/R)) ycoor=R*sin((90*PI/180)-(arclength/R)) midx = R*cos((90*PI/180)-(arclength/(2*R))) midy = R*sin((90*PI/180)-(arclength/(2*R))) fourthx =R*cos((90*PI/ 180)-(arclength/(4*R))) fourthy =R*sin((90*PI/180)-(arclength/(4*R))) fourthx2 =R*cos((90*PI/180) -((arclength*3)/(4*R))) fourthy2 =R*sin((90*PI/180 )-((arclength*3) /(4*R))) !**********Create Keypoints 1 throug 11: K( Point #, X-Coord, Y-Coord, ZCoord)**************** K,1, 0,0,0, K,2, 0,R,0 K,3, xcoor,ycoor,0 k,4, midx, midy,h2 K,5, midx,midy,(h2)+1 K,6, midx*(R-1)/R,midy*(R-1)/R,h2 !!!????? !K,7, midx-midy/R,midy+midx/R,h2 !K,7, midx*(R+1)/R,midy*(R+1)/R,h2 K,7, midx+cos(arclength/(2*R )),midy-sin(arclength/(2*R)),h2 k,8, fourthx,fourthy,h2/2 k,9, fourthx,fourthy,(h2/2)+1 k,10, fourthx*(R-1)/R ,fourthy*(R-1)/R,h2/2 !!!????? !k,11, fourthx-fourthy/ R,fourthy+f ourthx/R,h2/2 !k,11, fourthx*(R+1)/R ,fourthy*(R+1)/R,h2/2 k,11, fourthx+cos(arclength/(4*R )),fourthy-sin(arcl ength/(4*R)),h2/2 k,12, fourthx2,fourthy2,h2/2

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113 Appendix E: (continued) k,13, fourthx2,fourthy2,(h2/2)+1 k,14, fourthx2*(R-1)/R ,fourthy2*(R-1)/R,h2/2 !!!????? !k,15, fourthx2-fourthy2/ R,fourthy2+fourthx2/R,h2/2 !k,15, fourthx2*(R+1)/R ,fourthy2*(R+1)/R,h2/2 k,15, fourthx2+cos((arclength*3)/(4*R )),fourthy2-sin((arclength*3)/(4*R)),h2/2 !*********Create Beam using Lines a nd an Arc and divide into segments************************ LSTR, 4,5 !line 1 LSTR, 4,6 !line 2 Draws lines connecting keypoints LSTR, 4,7 !line 3 LSTR, 8,9 !line 4 LSTR, 8,10 !line 5 LSTR, 8,11 !line 6 LSTR, 12,13 !line 7 LSTR, 12,14 !line 8 LSTR, 12,15 !line 9 LESIZE, ALL,,,1 LARC, 2,8,1,R, !arc 10 LARC, 8,4,1,R, !arc 11 LARC, 4,12,1,R, !arc 12 LARC, 12,3,1,R, !arc 13 LESIZE, 10,,,30 LESIZE, 11,,,30 LESIZE, 12,,,30 LESIZE, 13,,,30 !***********MESH****************************************************** ********************** !rigid part skew axis!! real, 1 Use real constant set 1 type, 1 Use element type 1 mat, 1 use material property set 1 LMESH, 1,9 mesh lines 1-9

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114 Appendix E: (continued) !copliant arc! real, 2 Use real constant set 2 type, 1 Use element type 1 mat, 2 use material property set 2 LMESH, 10,13 mesh line 10,13 !******Get Node Numb ers at chosen keypoints************************************************** ksel,s,kp,,2 nslk,s *get,nkp2,node,0,num,max !Retriev es a value and stores it as a scalar parameter or part of an array parameter nsel,all ksel,all ksel,s,kp,,3 nslk,s *get,nkp3,node,0,num,max nsel,all ksel,all ksel,s,kp,,4 nslk,s *get,nkp4,node,0,num,max !Retriev es a value and stores it as a scalar parameter or part of an array parameter 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

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115 Appendix E: (continued) nslk,s *get,nkp7,node,0,num,max nsel,all ksel,all ksel,s,kp,,8 nslk,s *get,nkp8,node,0,num,max !Retriev es a value and stores it as a scalar parameter or part of an array parameter nsel,all ksel,all ksel,s,kp,,9 nslk,s *get,nkp9,node,0,num,max nsel,all ksel,all ksel,s,kp,,10 nslk,s *get,nkp10,node,0,num,max nsel,all ksel,all ksel,s,kp,,11 nslk,s *get,nkp11,node,0,num,max nsel,all ksel,all ksel,s,kp,,12 nslk,s *get,nkp12,node,0,num,max nsel,all ksel,all ksel,s,kp,,13 nslk,s *get,nkp13,node,0,num,max nsel,all ksel,all ksel,s,kp,,14

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116 Appendix E: (continued) nslk,s *get,nkp14,node,0,num,max nsel,all ksel,all ksel,s,kp,,15 nslk,s *get,nkp15,node,0,num,max nsel,all ksel,all FINISH !*********************************************************************** ********************** !********************** SOLUTION ************************************************************* !*********************************************************************** ********************** /SOL ANTYPE,0 Sp ecifies the analysis type and restart status and "0" means that it Performs a static analysis. Valid for all degrees of freedom NLGEOM,1 Includes large-deflection effects in a static or full transient analysis !CNVTOL,U,,0.000001,,0 !CNVTOL,F,,0.0001,,0 !Sets convergence va lues for nonlinear analyses SOLCONTROL,ON NEQIT,100 AUTOTS,ON !***************** Defines DOF constraints at keypoints**************************************************** DK,2, ,0, , ,UX,UY,UZ,ROTX,ROTY,ROTz DK,3, ,0, , ,UZ,ROTY,ROTX

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117 Appendix E: (continued) !*********************************************************************** ********* increments=100 loadsteps=70 *DO,step,1,loadsteps,1 theta=(step-1)*arcle ngth/(R*increments) !DDELE,12,ALL !*********************************************************************** ********* newx=R*cos((PI/2-(arclength/R-theta))) newy=R*sin((PI/2-(arclength/R-theta))) dispx=newx-xcoor dispy=newy-ycoor DK,3,UX,dispx DK,3,UY,dispy DK,3,ROTZ,theta LSWRITE,step *ENDDO LSSOLVE,1,loadsteps /STATUS,SOLU FINISH !*********************************************************************** ********** !********GET RESULTS ************************************************************* !*********************************************************************** ********** !**************************Displacements nodes 8,(9,10,11),4,(5,6,7,)3,2,12,(13,14,15)************************** /POST1 *DIM,rotx8,TABLE,loadSteps *DIM,roty8,TABLE,loadSteps *DIM,rotz8,TABLE,loadSteps

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121 Appendix E: (continued) *GET,disX,Node,nkp9,U,X *SET,disX9(nn),disX *GET,disY,Node,nkp9,U,Y *SET,disY9(nn),disY *GET,disz,Node,nkp9,U,Z *SET,disZ9(nn),disz *GET,rotx,Node,nkp10,ROT,X *SET,rotx10(nn),rotx *GET,roty,Node,nkp10,ROT,Y *SET,roty10(nn),roty *GET,rotz,Node,nkp10,ROT,Z *SET,rotz10(nn),rotz *GET,disX,Node,nkp10,U,X *SET,disX10(nn),disX *GET,disY,Node,nkp10,U,Y *SET,disY10(nn),disY *GET,disz,Node,nkp10,U,Z *SET,disZ10(nn),disz *GET,rotx,Node,nkp11,ROT,X *SET,rotx11(nn),rotx *GET,roty,Node,nkp11,ROT,Y *SET,roty11(nn),roty *GET,rotz,Node,nkp11,ROT,Z *SET,rotz11(nn),rotz *GET,disX,Node,nkp11,U,X *SET,disX11(nn),disX *GET,disY,Node,nkp11,U,Y *SET,disY11(nn),disY *GET,disz,Node,nkp11,U,Z *SET,disZ11(nn),disz *GET,rotx,Node,nkp4,ROT,X *SET,rotx4(nn),rotx *GET,roty,Node,nkp4,ROT,Y *SET,roty4(nn),roty *GET,rotz,Node,nkp4,ROT,Z *SET,rotz4(nn),rotz

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122 Appendix E: (continued) *GET,disX,Node,nkp4,U,X *SET,disX4(nn),disX *GET,disY,Node,nkp4,U,Y *SET,disY4(nn),disY *GET,disz,Node,nkp4,U,Z *SET,disZ4(nn),disz *GET,rotx,Node,nkp5,ROT,X *SET,rotx5(nn),rotx *GET,roty,Node,nkp5,ROT,Y *SET,roty5(nn),roty *GET,rotz,Node,nkp5,ROT,Z *SET,rotz5(nn),rotz *GET,disX,Node,nkp5,U,X *SET,disX5(nn),disX *GET,disY,Node,nkp5,U,Y *SET,disY5(nn),disY *GET,disz,Node,nkp5,U,Z *SET,disZ5(nn),disz *GET,rotx,Node,nkp6,ROT,X *SET,rotx6(nn),rotx *GET,roty,Node,nkp6,ROT,Y *SET,roty6(nn),roty *GET,rotz,Node,nkp6,ROT,Z *SET,rotz6(nn),rotz *GET,disX,Node,nkp6,U,X *SET,disX6(nn),disX *GET,disY,Node,nkp6,U,Y *SET,disY6(nn),disY *GET,disz,Node,nkp6,U,Z *SET,disZ6(nn),disz *GET,rotx,Node,nkp7,ROT,X *SET,rotx7(nn),rotx *GET,roty,Node,nkp7,ROT,Y *SET,roty7(nn),roty *GET,rotz,Node,nkp7,ROT,Z *SET,rotz7(nn),rotz *GET,disX,Node,nkp7,U,X

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123 Appendix E: (continued) *SET,disX7(nn),disX *GET,disY,Node,nkp7,U,Y *SET,disY7(nn),disY *GET,disz,Node,nkp7,U,Z *SET,disZ7(nn),disz !Point 3???? *GET,rotz,Node,nkp3,ROT,Z *SET,rotz3(nn),rotz *GET,disX,Node,nkp3,U,X *SET,disX3(nn),disX *GET,disY,Node,nkp3,U,Y *SET,disY3(nn),disY *GET,rotx,Node,nkp12,ROT,X *SET,rotx12(nn),rotx *GET,roty,Node,nkp12,ROT,Y *SET,roty12(nn),roty *GET,rotz,Node,nkp12,ROT,Z *SET,rotz12(nn),rotz *GET,disX,Node,nkp12,U,X *SET,disX12(nn),disX *GET,disY,Node,nkp12,U,Y *SET,disY12(nn),disY *GET,disz,Node,nkp12,U,Z *SET,disZ12(nn),disz *GET,rotx,Node,nkp13,ROT,X *SET,rotx13(nn),rotx *GET,roty,Node,nkp13,ROT,Y *SET,roty13(nn),roty *GET,rotz,Node,nkp13,ROT,Z *SET,rotz13(nn),rotz *GET,disX,Node,nkp13,U,X *SET,disX13(nn),disX *GET,disY,Node,nkp13,U,Y *SET,disY13(nn),disY *GET,disz,Node,nkp13,U,Z

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124 Appendix E: (continued) *SET,disZ13(nn),disz *GET,rotx,Node,nkp14,ROT,X *SET,rotx14(nn),rotx *GET,roty,Node,nkp14,ROT,Y *SET,roty14(nn),roty *GET,rotz,Node,nkp14,ROT,Z *SET,rotz14(nn),rotz *GET,disX,Node,nkp14,U,X *SET,disX14(nn),disX *GET,disY,Node,nkp14,U,Y *SET,disY14(nn),disY *GET,disz,Node,nkp14,U,Z *SET,disZ14(nn),disz *GET,rotx,Node,nkp15,ROT,X *SET,rotx15(nn),rotx *GET,roty,Node,nkp15,ROT,Y *SET,roty15(nn),roty *GET,rotz,Node,nkp15,ROT,Z *SET,rotz15(nn),rotz *GET,disX,Node,nkp15,U,X *SET,disX15(nn),disX *GET,disY,Node,nkp15,U,Y *SET,disY15(nn),disY *GET,disz,Node,nkp15,U,Z *SET,disZ15(nn),disz !**************************Reactions forces node 2******************************* *GET,momx,Node,nkp2,RF,MX *SET,momx2(nn),momx *GET,fx,Node,nkp2,RF,FX *SET,fx2(nn),fx *GET,momy,Node,nkp2,RF,MY *SET,momy2(nn),momy *GET,fy,Node,nkp2,RF,FY

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125 Appendix E: (continued) *SET,fy2(nn),fy *GET,momz,Node,nkp2,RF,MZ *SET,momz2(nn),momz *GET,fz,Node,nkp2,RF,FZ *SET,fz2(nn),fz !**************************Reactions forces node 3******************************* *GET,momx,Node,nkp3,RF,MX *SET,momx3(nn),momx *GET,fx,Node,nkp3,RF,FX *SET,fx3(nn),fx *GET,momy,Node,nkp3,RF,MY *SET,momy3(nn),momy *GET,fy,Node,nkp3,RF,FY *SET,fy3(nn),fy *GET,momz,Node,nkp3,RF,MZ *SET,momz3(nn),momz *GET,fz,Node,nkp3,RF,FZ *SET,fz3(nn),fz *ENDDO /output,curvebeamarcsdevic e%LAMBDAdg%,txt,,Append !*********************************************************************** *************** !***************FILE HEADER: BEAM DATA************************************************* !*********************************************************************** *************** *MSG,INFO,'t','w','R','E','arclength' %-8C %-8C %-8C %-8C %-8C *VWRITE,h2,b2,R,E2,arclength %16.8G %-16.8G %-16.8G %-16.8G %-16.8G !*********************************************************************** ************

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126 Appendix E: (continued) !**************DISPLACEMENT DATA SET************************************************ !*********************************************************************** ************ *MSG,INFO,'rotX8','rotY8','rotZ8','disX8','disY8','disZ8' %-8C %-8C %-8C %-8C %-8C %-8C *VWRITE,rotx8(1),roty8(1),rotz8( 1),disX8(1),disY8(1),disZ8(1) %16.8G %-16.8G %-16.8G %16.8G %-16.8G %-16.8G *MSG,INFO,'rotX9','rotY9','rotZ9','disX9','disY9','disZ9' %-8C %-8C %-8C %-8C %-8C %-8C *VWRITE,rotx9(1),roty9(1),rotz9( 1),disX9(1),disY9(1),disZ9(1) %16.8G %-16.8G %-16.8G %16.8G %-16.8G %-16.8G *MSG,INFO,'rotX10','rotY10','rotZ 10','disX10','disY10','disZ10' %-8C %-8C %-8C %-8C %-8C %-8C *VWRITE,rotx10(1),roty10( 1),rotz10(1),disX10(1) ,disY10(1),disZ10(1) %16.8G %-16.8G %-16.8G %16.8G %-16.8G %-16.8G *MSG,INFO,'rotX11','rotY11','rotZ 11','disX11','disY11','disZ11' %-8C %-8C %-8C %-8C %-8C %-8C *VWRITE,rotx11(1),roty11( 1),rotz11(1),disX11(1) ,disY11(1),disZ11(1) %16.8G %-16.8G %-16.8G %16.8G %-16.8G %-16.8G *MSG,INFO,'rotX12','rotY12','rotZ 12','disX12','disY12','disZ12' %-8C %-8C %-8C %-8C %-8C %-8C *VWRITE,rotx12(1),roty12( 1),rotz12(1),disX12(1) ,disY12(1),disZ12(1) %16.8G %-16.8G %-16.8G %16.8G %-16.8G %-16.8G *MSG,INFO,'rotX13','rotY13','rotZ 13','disX13','disY13','disZ13'

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127 Appendix E: (continued) %-8C %-8C %-8C %-8C %-8C %-8C *VWRITE,rotx13(1),roty13( 1),rotz13(1),disX13(1) ,disY13(1),disZ13(1) %16.8G %-16.8G %-16.8G %16.8G %-16.8G %-16.8G *MSG,INFO,'rotX14','rotY14','rotZ 14','disX14','disY14','disZ14' %-8C %-8C %-8C %-8C %-8C %-8C *VWRITE,rotx14(1),roty14( 1),rotz14(1),disX14(1) ,disY14(1),disZ14(1) %16.8G %-16.8G %-16.8G %16.8G %-16.8G %-16.8G *MSG,INFO,'rotX15','rotY15','rotZ 15','disX15','disY15','disZ15' %-8C %-8C %-8C %-8C %-8C %-8C *VWRITE,rotx15(1),roty15( 1),rotz15(1),disX15(1) ,disY15(1),disZ15(1) %16.8G %-16.8G %-16.8G %16.8G %-16.8G %-16.8G *MSG,INFO,'rotX4','rotY4','rotZ4','disX4','disY4','disZ4' %-8C %-8C %-8C %-8C %-8C %-8C *VWRITE,rotx4(1),roty4(1),rotz4( 1),disX4(1),disY4(1),disZ4(1) %16.8G %-16.8G %-16.8G %16.8G %-16.8G %-16.8G *MSG,INFO,'rotX5','rotY5','rotZ5','disX5','disY5','disZ5' %-8C %-8C %-8C %-8C %-8C %-8C *VWRITE,rotx5(1),roty5(1),rotz5( 1),disX5(1),disY5(1),disZ5(1) %16.8G %-16.8G %-16.8G %16.8G %-16.8G %-16.8G *MSG,INFO,'rotX6','rotY6','rotZ6','disX6','disY6','disZ6' %-8C %-8C %-8C %-8C %-8C %-8C *VWRITE,rotx6(1),roty6(1),rotz6( 1),disX6(1),disY6(1),disZ6(1) %16.8G %-16.8G %-16.8G %16.8G %-16.8G %-16.8G *MSG,INFO,'rotX7','rotY7','rotZ7','disX7','disY7','disZ7' %-8C %-8C %-8C %-8C %-8C %-8C

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128 Appendix E: (continued) *VWRITE,rotx7(1),roty7(1),rotz7( 1),disX7(1),disY7(1),disZ7(1) %16.8G %-16.8G %-16.8G %16.8G %-16.8G %-16.8G !!POINTS 3 AND 2????? *MSG,INFO,'rotZ3','disX3','disY3' %-8C %-8C %-8C *VWRITE,rotz3(1),disX3(1),disY3(1) %16.8G %-16.8G %-16.8G !*********************************************************************** ************ !**************REACTIONS AT NODE 3************************************************** !*********************************************************************** ************ *MSG,INFO,'momx3','momy3', 'momz3','fx3', 'fy3','fz3' %-8C %-8C %-8C %-8C %-8C %-8C *VWRITE,momx3(1),momy3(1),mom z3(1),fx3(1),fy3(1),fz3(1) %16.8G %-16.8G %-16.8G %16.8G %-16.8G %-16.8G !*********************************************************************** ************ !**************REACTIONS AT NODE 2************************************************** !*********************************************************************** ************ *MSG,INFO,'momx2','momy2', 'momz2','fx2', 'fy2','fz2' %-8C %-8C %-8C %-8C %-8C %-8C *VWRITE,momx2(1),momy2(1),mom z2(1),fx2(1),fy2(1),fz2(1) %16.8G %-16.8G %-16.8G %16.8G %-16.8G %-16.8G

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129 Appendix E: (continued) !*********************************************************************** ************ !*********************************************************************** ************ !*********************************************************************** ************ /output FINISH *ENDDO *ENDDO

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132 Appendix F: (continued) rotx11 = DATA4(:,1); roty11 = DATA4(:,1); rotz11 = DATA4(:,3); disx11 = DATA4(:,4); disy11 = DATA4(:,5); disz11 = DATA4(:,6); rotx12 = DATA5(:,1); roty12 = DATA5(:,1); rotz12 = DATA5(:,3); disx12 = DATA5(:,4); disy12 = DATA5(:,5); disz12 = DATA5(:,6); rotx13 = DATA6(:,1); roty13 = DATA6(:,1); rotz13 = DATA6(:,3); disx13 = DATA6(:,4); disy13 = DATA6(:,5); disz13 = DATA6(:,6); rotx14 = DATA7(:,1); roty14 = DATA7(:,1); rotz14 = DATA7(:,3); disx14 = DATA7(:,4); disy14 = DATA7(:,5); disz14 = DATA7(:,6); rotx15 = DATA8(:,1); roty15 = DATA8(:,1); rotz15 = DATA8(:,3); disx15 = DATA8(:,4); disy15 = DATA8(:,5); disz15 = DATA8(:,6); rotx4 = DATA9(:,1); roty4 = DATA9(:,1); rotz4 = DATA9(:,3); disx4 = DATA9(:,4); disy4 = DATA9(:,5); disz4 = DATA9(:,6); rotx5 = DATA10(:,1); roty5 = DATA10(:,1); rotz5 = DATA10(:,3); disx5 = DATA10(:,4); disy5 = DATA10(:,5); disz5 = DATA10(:,6);

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133 Appendix F: (continued) rotx6 = DATA11(:,1); roty6 = DATA11(:,1); rotz6 = DATA11(:,3); disx6 = DATA11(:,4); disy6 = DATA11(:,5); disz6 = DATA11(:,6); rotx7 = DATA12(:,1); roty7 = DATA12(:,1); rotz7 = DATA12(:,3); disx7 = DATA12(:,4); disy7 = DATA12(:,5); disz7 = DATA12(:,6); rotZ3 = DATA13(:,1); disx3 = DATA13(:,2); disy3 = DATA13(:,3); momx3 = DATA14(:,1); momy3 = DATA14(:,2); momz3 = DATA14(:,3); fx3 = DATA14(:,4); fy3 = DATA14(:,5); fz3 = DATA14(:,6); momx2 = DATA15(:,1); momy2 = DATA15(:,2); momz2 = DATA15(:,3); fx2 = DATA15(:,4); fy2 = DATA15(:,5); fz2 = DATA15(:,6); arclength = DATA16(:,5); R = DATA16(:,3); h2 = DATA16(:,1); x2 = zeros(70,1); y2 = zeros(70,1); z2 = zeros(70,1); x3 = R*sin(arclength/R)+disx3; y3 = R*cos(arclength/R)+disy3; z3 = zeros(70,1); R3 = sqrt(x3.^2+y3.^2); x4 = R*sin(arclength/(2*R))+disx4; y4 = R*cos(arclength/(2*R))+disy4; z4 = 0+disz4; R4 = sqrt(x4.^2+y4.^2+z4.^2);

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134 Appendix F: (continued) x8 = R*sin(arclength/(4*R))+disx8; y8 = R*cos(arclength/(4*R))+disy8; z8 = h2/2+disz8; R8 = sqrt(x8.^2+y8.^2+z8.^2); x12 = R*sin((arclength*3)/(4*R))+disx12; y12 = R*cos((arclength*3)/(4*R))+disy12; z12 = h2/2+disz12; R12 = sqrt(x12.^2+y12.^2+z12.^2); x13 = R*sin((arclength*3)/(4*R))+disx13; y13 = R*cos((arclength*3)/(4*R))+disy13; z13 = h2/2+1+disz13; x14 = R*sin((arclength*3)/(4*R))*(R-1)/R+disx14; y14 = R*cos((arclength*3)/(4*R))*(R-1)/R+disy14; z14 = h2/2+disz14; x15 = R*sin((arclength*3)/(4*R))+cos((arclength*3)/(4*R))+disx15; y15 = R*cos((arclength*3)/(4*R))-sin((arclength*3)/(4*R))+disy15; z15 = h2/2+disz15; %%%thetas with number are moving on plane phis with number are vertical %%%angle out plane theta3=atan2(y3,x3); theta8=atan2(y8,x8); phi8=atan2(z8,sqrt(x8.^2+y8.^2)); theta12=atan2(y12,x12); phi12=atan2(z12,sqrt(x12.^2+y12.^2)); theta4=atan2(y4,x4); phi4=atan2(z4,sqrt(x4.^2+y4.^2)); Dx = [x15-x12 y15-y12 z15-z12]'; %Stick original parallel with Cx Dz = [x13-x12 y13-y12 z13-z12]'; %Stick original parallel with Cz %Dy= [x14-x12 y14-y12 z14-z12]'; %Stick original antiparallel with Cy Dy = [x12-x14 y12-y14 z12-z14]'; %use opposite of stick direction A = eye(3); %identity Matrix for i = 1:length(rotz12) R_B = [ cos( theta12(i)-(pi/2)) -sin(theta12(i)-(pi/2)) 0 ; sin(theta12(i)-(pi/2)) cos( theta12(i)-(pi/2)) 0 ; 0 0 1]; %%%%Beta out plane angle of point 12 = phi 12 BETA = atan2(z12,sqrt(x12.^2+y12.^2));

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135 Appendix F: (continued) R_C = [1 0 0;0 cos(BETA(i)) -sin(BETA(i)) ;0 sin(BETA(i)) cos(BETA(i))]; Bx(:,i) = R_C*A(:,1); By(:,i) = R_C*A(:,2); Bz(:,i) = R_C*A(:,3); Cx(:,i) = R_B*Bx(:,i); Cy(:,i) = R_B*By(:,i); Cz(:,i) = R_B*Bz(:,i); RR = [Dx(:,i) Dy(:,i) Dz(:,i)]*inv([Cx(:,i) Cy(:,i) Cz(:,i)]); ROT(i,:,:) = RR; v1(i)=RR(1,2)*RR(2,3)-(RR(2,2)-1)*RR(1,3); v2(i)=RR(2,1)*RR(1,3)-(RR(1,1)-1)*RR(2,3); v3(i)=(RR(1,1)-1)*(RR(2,2)-1)-(RR(1,2)*RR(2,3)); V=[v1(i) v2(i) v3(i)]; TRC=trace(RR); CDROTMAG(i)=acos((TRC-1)/2); % Acording to plannar case CDROTMAG=theta0 end %Shows Position of the axis of rotation with respec to point 12. figure(1) quiver3(x12,y12,z12,v1,v2,v3) %plot(rotZ3,[x12,y12-100,z12,v1',v2',v3']) %Shows Linear relationship between Theta0 and RotZ3 figure(2) clf hold on plot(rotZ3,CDROTMAG) %If Cy and Dy are matching then RR is a rotacion about the Cy figure(3) clf hold on %quiver3(x12,y12,z12,v1,v2,v3) plot(rotZ3,Cy(1,:)) plot(rotZ3,Cy(2,:), 'g' ) plot(rotZ3,Cy(3,:), 'r' ) plot(rotZ3,Dy(1,:), 'c' )

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136 Appendix F: (continued) plot(rotZ3,Dy(2,:), 'm' ) plot(rotZ3,Dy(3,:), 'y' ) plot(rotZ3,x12/R, 'c*' ) plot(rotZ3,y12/R, 'm*' ) plot(rotZ3,z12/R, 'y*' ) % Cy -Radial, Cz longitude, Cx latitude, Psi-angel b/w C and D Psiy= acos(dot(Cy,Dy)./(sqrt(dot(Cy,Cy)).*sqrt(dot(Dy,Dy))))*180/pi; Psix= real(acos(dot(Cx,Dx)./(sqrt(dot(Cx,Cx)).*sqrt(dot(Dx,Dx)))))*180/pi; Psiz= acos(dot(Cz,Dz)./(sqrt(dot(Cz,Cz)).*sqrt(dot(Dz,Dz))))*180/pi; rot_val(:,1) = real(acos((dot(Cx,Dx)./(sqrt(dot(Cx,Cx)).*sqrt(dot(Dx,Dx))))))*180/pi; rot_val(:,2) = asin((dot(Cx,Dz)./(sqrt(dot(Cx,Cx)).*sqrt(dot(Dx,Dx)))))*180/pi; rot_val(:,3) = acos((dot(Cz,Dz)./(sqrt(dot(Cx,Cx)).*sqrt(dot(Dx,Dx)))))*180/pi; rot_val(:,4) = asin((dot(Cz,Dx)./(sqrt(dot(Cx,Cx)).*sqrt(dot(Dx,Dx)))))*180/pi; %Psi-angel b/w C and D-b/w cy and dy must be zero because radial direction no twist! figure(4) clf hold on plot(Psiy, 'r-*' ) plot(Psix, 'b-*' ) plot(Psiz, 'g-*' ) % Checks radial displacement of keypoints figure(5) clf hold on plot(R4/R, 'r' ) plot(R8/R, 'b' ) plot(R12/R, 'm' ) plot(R3/R, 'k' ) %normal vector to Cy, and Dy, X,Z CDY=cross(Cy,Dy)./([1;1;1]*(sin(Psiy*pi/180).*(sqrt(dot(Cy,Cy)).*sqrt(d ot(Dy,Dy))))); CDX=cross(Cx,Dx); CDZ=cross(Cz,Dz); %vectors from center of sphere to point 12 must match Cy and Dy figure(6) clf hold on quiver3(zeros(1,70),zeros(1,70),zeros(1,70),(x12)/R,(y12)/R,(z12)/R, 'b' )

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137 Appendix F: (continued) quiver3(zeros(1,70),zeros(1,70),zeros(1,70),Cy(1,:),Cy(2,:),Cy(3,:), 'r' ) quiver3(zeros(1,70),zeros(1,70),zeros(1,70),Dy(1,:),Dy(2,:),Dy(3,:), 'g' ) view(3) grid on % checks quarter versus half symmetry for HKP leg, perfect half symetry not % perfect quarter symetry error_theta8 =(90-theta8*180/pi)*4 (90-theta3*180/pi); error_theta4 =(90-theta4*180/pi)*2 (90-theta3*180/pi); error_theta12 =(90-theta12*180/pi)*4/3 (90-theta3*180/pi); figure(7) clf hold on plot(error_theta12, 'r' ); plot(error_theta4, 'b' ); plot(error_theta8, 'k' ); % X axis will be captheta ylabel( 'error (deg)' ) error1=abs(theta4-theta8); error2=abs(theta12-theta4); error3=abs(theta8-pi/2); error4=abs(theta3-theta12); checcc=error1-error2 checcc2=error3-error4 %Shows errors betwenn half su=ymety and quater symetry thetas figure(8) clf hold on plot([1:70],[error1, error2, error3, error4 ]) figure(81) clf hold on plot([1:70],[ error3, error4 ]) % checks quarter versus half symmetry for HKP leg error_phia = phi8*180/pi*2 phi4*180/pi; error_phib = phi8*180/pi phi12*180/pi; error_phic =phi4*180/pi phi12*180/pi*2; error5=abs(phi4-phi8); error6=abs(phi12-phi4);

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138 Appendix F: (continued) error7=phi8; error8=phi12; %Shows errors betwenn half su=ymety and quater symetry phis figure(9) clf hold on plot([1:70],[error5 error6 error7 error8]); % checks quarter versus half symmetry for HKP leg figure(10) clf hold on plot(error_phia, 'r' ); plot(error_phib, 'b' ); plot(error_phic, 'k' ); % X axis will be captheta ylabel( 'error (deg)' ) %Show the traagectory of points 12 4 and 8 figure(11) clf hold on plot3(x8,y8,z8, 'r*' ,x4,y4,z4, 'b*' ,x12,y12,z12, 'g*' ); grid on axis equal view(3) scale=1/8; %Shows that most of the force is apply in the x direction the y and z %componets of the force are close to 0 figure(12) clf hold on plot(sqrt(fz3.^2+fx3.^2+fy3.^2)) hold on plot(fx3, 'r' ) plot(fy3, 'g' ) plot(fz3, 'k' ) grid on %Shows the reaction forces vectors at the fix point 2 figure(13) quiver3(x2,y2,z2,fx2,fy2,fz2) view(3) hold on %Show that mom x and y are equal and apposite at points 2 and 3 figure(14) clf hold on

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139 Appendix F: (continued) quiver(x2,y2,momx3,momy3,0, 'b.' ) quiver(x2,y2,momx2,momy2,0, 'r.' ) view(3) hold on grid on %Shows the moment vector behavior on point 3 as it moves figure(15) quiver3(x3,y3,z3,momx3,momy3,momz3,0) view(3) hold on %beta and alpha are radians % %beta = phi8; % %alpha= pi/2-theta8 % % %l = 100; % %a= l+disx8; % %b =disz8; %PHI =arclength/R alpha %beta and alpha are radians %CHANGED DEFFINITION OF ALPHA AND BETA!!!! alpha= (theta4-theta3)/2; %%%change of theta of curved beam center %%%% beta lower case half of out plane angle of center of beam = BETA of %%%% point 12 because of symetry there are some error between half symetry %%%% and quater symetry!!! beta = phi4/2; PHI = alpha(1)-alpha; %not vertical phi change in alpha!!! %%%%%%%DEFINING l,a,b,gamma,Captheta,theta0,torque%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% gamma = 4*R/arclength*atan2(1-cos(beta).*cos(PHI),cos(beta).*sin(PHI)); %planar gamma = -(a.^2 2*a*l+l^2+b.^2)./(2*a*l-2*l^2); theta = atan2(tan(beta),sin((gamma*arclength/(4*R))-PHI)); %atan2(b,a-(1-gamma)*l);

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140 Appendix F: (continued) lambda=arclength/(R*4); % Phi=atan2(1cos(theta),cot(gamma*lambda)+(cos(theta).*tan(gamma*lambda))) % Beta=asin(sin(gamma*lambda).*sin(theta)) % Alpha=lambda-Phi % figure(12) % clf % hold on % plot(PHI,alpha,'b*',Phi,alpha,'b') % plot(beta,alpha,'g*',Beta,alpha,'g') % plot(alpha,alpha,'r*',Alpha,alpha,'r') % rotation of point 3---to find M12 r=[x12-x3 y12-y3 z12-z3]; M12 = cross(r,[fx3 fy3 fz3]); mom12=[momx3 momy3 momz3]-M12; momx12=mom12(:,1); momy12=mom12(:,2); momz12=mom12(:,3); Magmom12=sqrt(momx12.^2+momy12.^2+momz12.^2); MAXMagmom12=max(Magmom12); M12=Magmom12/MAXMagmom12; %Compares the behavior of the reaction moment vectors at point 12 and 3 as %the mechanism moves figure(16) clf hold on quiver3(x12,y12,z12,momx12,momy12,momz12) quiver3(x3,y3,z3,momx3,momy3,momz3, 'r' ) view(3) M_Mag = sqrt(momx3.^2+momy3.^2+momz3.^2); M_Mag_max= max(M_Mag); % M_theta = atan2(momy3,momx3) % M_phi = atan2(momz3,sqrt(momy3.^2+momx3.^2)) %Shows most of the reaction force at 3 happens on the x direction, %most of the reaction moment at 3 ias about the y

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141 Appendix F: (continued) % and the moments at 12 are close to zero %momz12 represent twist about z axis must be close to zero figure(17) clf hold on plot(sqrt(fz3.^2+fx3.^2+fy3.^2)) plot(sqrt(momx12.^2+momy12.^2+momz12.^2), '--' ) plot(sqrt(momx3.^2+momy3.^2+momz3.^2), '*' ) hold on plot(fx3, 'r' ) plot(fy3, 'g' ) plot(fz3, 'k' ) plot(momx12, 'r--' ) plot(momy12, 'g--' ) plot(momz12, 'k--' ) plot(momx3, 'r*' ) plot(momy3, 'g*' ) plot(momz3, 'k*' ) theta(1) = 0; theta0 = CDROTMAG; %Compares behavior of point 3 and 12 figure(18) clf hold on plot3(x12,y12,z12, '*' ,x3,y3,0*x3, '*' ) view(3) theta(1) =0; theta0 = CDROTMAG; E = 169000; b2 =arclength/20; I = (b2*h2^3)/12; M = (M_Mag*arclength^2)/(E*I); % Nondimensionalization M_max = max(M); %Scales moment between 0 and 1 M = M/M_max; R1 = normrnd(0.1,.1,[size(M)]);

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142 Appendix F: (continued) X = [ones(size(M(2:end))) 1./M(2:end) 1./M(2:end).^2 1.*M(2:end) 1.*M(2:end).^2]; Y = gamma(2:end); B = inv(X'*X)*X'*Y; %Gamma = 1./(B(1)+B(2)./M+B(3)./M.^2+B(4)./M.^3+B(5)./M.^4+B(6)./M.^5); Gamma = B(1)+B(2)./M+B(3)./M.^2 +B(4).*M +B(5).*M.^2; Csgamma=POLYFIT(M*M_max,gamma,0); %Theta = atan2(b,a-(1-Gamma)*l); Theta = atan2(tan(beta),sin((Gamma*arclength/(4*R))-PHI)); %ThetaC= atan2(b,a-(1-.75)*l); ThetaC = atan2(tan(beta),sin(( Csgamma*arclength/(4*R))-PHI)); %gives aproximation constant Ctheta (theta0=(Ctheta)*CAPtheta Cstheta=POLYFIT(Theta(2:end),(theta0(2:end)./Theta(2:end)')',0); figure(19) clf hold on plot(Theta(2:end)*180/pi,theta0(2:end)./Theta(2:end)', 'b' ,Theta(2:end)* 180/pi,Cstheta.*ones(69,1)', 'r' ) %OJO FOR THIS ARC 105%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Ctheta=1.3427%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%Shows the aproximation of theta0=Ctheta*Theta%%%%%%%% figure(20) clf hold on plot(Theta(2:end)*180/pi,theta0(2:end)*180/pi, 'r' ) plot(Theta(2:end)*180/pi,Theta(2:end)*Cstheta*180/pi, 'b' ) epsilon = gamma(2:end)-Gamma(2:end); SST = sum((gamma(2:end)-mean(gamma(2:end))).^2); SSE = sum(epsilon.^2);

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143 Appendix F: (continued) s_squared = SSE/(length(gamma(2:end))-length(B)); s = sqrt(s_squared); rsqrd = 1SSE/SST; Var_b = inv(X'*X)*s_squared; X_ortho = X'*X; [V,D] = eig(X_ortho); X2(:,1) = X*V(:,1); X2(:,2) = X*V(:,2); X2(:,3) = X*V(:,3); X2(:,4) = X*V(:,4); X2(:,5) = X*V(:,5); C = inv(X2'*X2)*X2'*Y; %Gamma2 = C(1)*X2(:,1)+C(2)*X2(:,2)+C(3)*X2(:,3)+C(4)*X2(:,4)+C(5)*X2(:,5); Gamma2 = C(1)*X2(:,1)+C(2)*X2(:,2)+C(3)*X2(:,3)+C(4)*X2(:,4)+C(5)*X2(:,5); %Theta2 = atan2(b(2:end),a(2:end)-(1-Gamma2)*l); epsilon2 = gamma(2:end)-Gamma2; SSE2 = sum(epsilon2.^2); s_squared2 = SSE2/(length(gamma(2:end))-length(C)); s2 = sqrt(s_squared2); rsqrd2 = 1SSE2/SST Var_c = inv(X2'*X2)*s_squared2; B1_prime = C(1)*V(1,1)+C(2)*V(1,2)+C(3)*V(1,3)+C(4)*V(1,4)+C(5)*V(1,5); % compare with B(1) B2_prime = C(1)*V(2,1)+C(2)*V(2,2)+C(3)*V(2,3)+C(4)*V(2,4)+C(5)*V(2,5); % compare with B(2) B3_prime = C(1)*V(3,1)+C(2)*V(3,2)+C(3)*V(3,3)+C(4)*V(3,4)+C(5)*V(3,5); % compare with B(3) B4_prime = C(1)*V(4,1)+C(2)*V(4,2)+C(3)*V(4,3)+C(4)*V(4,4)+C(5)*V(4,5); % compare with B(4) B5_prime = C(1)*V(5,1)+C(2)*V(5,2)+C(3)*V(5,3)+C(4)*V(5,4)+C(5)*V(5,5); % compare with B(5) B_var(1) = abs(sqrt(Var_c(1,1))*V(1,1)+sqrt(Var_c(2,2))*V(1,2)+sqrt(Var_c(3,3))*V( 1,3)+sqrt(Var_c(4,4))*V(1,4)+sqrt(Var_c(5,5))*V(1,5));

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144 Appendix F: (continued) B_var(2) = abs(sqrt(Var_c(1,1))*V(2,1)+sqrt(Var_c(2,2))*V(2,2)+sqrt(Var_c(3,3))*V( 2,3)+sqrt(Var_c(4,4))*V(2,4)+sqrt(Var_c(5,5))*V(2,5)); B_var(3) = abs(sqrt(Var_c(1,1))*V(3,1)+sqrt(Var_c(2,2))*V(3,2)+sqrt(Var_c(3,3))*V( 3,3)+sqrt(Var_c(4,4))*V(3,4)+sqrt(Var_c(5,5))*V(3,5)); B_var(4) = abs(sqrt(Var_c(1,1))*V(4,1)+sqrt(Var_c(2,2))*V(4,2)+sqrt(Var_c(3,3))*V( 4,3)+sqrt(Var_c(4,4))*V(4,4)+sqrt(Var_c(5,5))*V(4,5)); B_var(5) = abs(sqrt(Var_c(1,1))*V(5,1)+sqrt(Var_c(2,2))*V(5,2)+sqrt(Var_c(3,3))*V( 5,3)+sqrt(Var_c(4,4))*V(5,4)+sqrt(Var_c(5,5))*V(5,5)); B_std = sqrt(B_var); % 95 % 2 sided confidence interval ie mean + or interval t_statistic = tinv(.975,length(Y)-length(B)); CI = t_statistic*B_std Gamma_minus = B(1)-CI(1)+(B(2)-CI(2))./M+(B(3)-CI(3))./M.^2 +(B(4)CI(4)).*M +(B(5)-CI(5)).*M.^2; Gamma_plus = B(1)+CI(1)+(B(2)+CI(2))./M+(B(3)+CI(3))./M.^2 +(B(4)+CI(4)).*M +(B(5)+CI(5)).*M.^2; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure(41) clf plot(Gamma(2:end),M(2:end)*M_max, 'b' Gamma_minus(2:end),M(2:end)*M_max, 'r' Gamma_plus(2:end),M(2:end)*M_max, 'g' ); %%%%%%Calculations Using model lambda=arclength/(R*4); Phi=atan2(1cos(Theta),cot(Gamma*lambda)+(cos(Theta).*tan(Gamma*lambda))); Beta=asin(sin(Gamma*lambda).*sin(Theta)); Alpha=lambda-Phi; %Shows that the equality of Phi, Beta and Alpha when found using gamma vs. %the ones obtain using the data guiven by ansys figure(21) clf hold on plot(PHI(2:end),alpha(2:end), 'b*' ,Phi(2:end),alpha(2:end), 'k' ) plot(beta(2:end),alpha(2:end), 'g*' ,Beta(2:end),alpha(2:end), 'k' ) plot(alpha(2:end),alpha(2:end), 'r*' ,Alpha(2:end),alpha(2:end), 'k' )

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145 Appendix F: (continued) %Show range of gammas figure(22) plot([Gamma(2:end) Gamma_minus(2:end) Gamma_plus(2:end)]) %%%%%%%%%%%FIGURE 1 alpha vs beta %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure(23); clf plot(alpha,beta, 'b*' Alpha(2:end),Beta(2:end), 'r' ,(lambda-atan2(1cos(ThetaC),cot( Csgamma*lambda)+(cos(ThetaC).*tan( Csgamma*lambda)))),asin(sin( Csgamma*lambda).*sin(ThetaC)), 'g-' ); %%%%%%%%%%%FIGURE 2 M vs theta %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure(24); clf plot(M,theta*180/pi, 'b*' ,M(2:end),Theta(2:end)*180/pi, 'R' ,M(2:end),Thet aC(2:end)*180/pi, 'G-' ); %%%%%%%%%%%FIGURE 3 M vs gamma %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure(25); clf plot(gamma,M*M_max, 'b*' ,Gamma(2:end),M(2:end)*M_max, 'R' Csgamma,M*M_max, 'G*' ); % G3c = ylabel('\gamma'); % set(G3c,'Rotation',0,'fontsize' ,12) % mytexstr = '$\frac{M l^2}{EI}$'; % Gc= xlabel(mytexstr,'interpreter','latex','fontsize',12,'units','norm'); G4c = legend( 'Data' '\gamma*' '\gamma' ); % hold on %%%%%%%%%%%ERROR %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% erroro = (alpha Alpha).^2 + (beta Beta).^2;

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146 Appendix F: (continued) %erroro = ( a/l ((1-Gamma)+Gamma.*cos(Theta))).^2 + (b/l Gamma.*sin(Theta)).^2; %disp_mag = ((((1-Gamma)+Gamma.*cos(Theta))a(round(end/3))/l).^2+(Gamma.*sin(Theta)-b(round(end/3))/l).^2).^.5; total_erroro = trapz(M(2:end),erroro(2:end)); %+trapz(Gamma(2:end),erroro(2:end)); %total_erroro = sqrt(max(erroro(2:end)./diff(M))) %%%Define tangent force get rid of radial component does not do work n_r3 = [x3 y3]./(sqrt(x3.^2+y3.^2)*[1 1]); F3 = [fx3 fy3]; F_r3 = (dot(F3',n_r3')'*[1 1]).*n_r3; Ft_3=F3-F_r3; q=Ft_3(1:70)'; p=Ft_3(71:140)'; Ftan=sqrt(q.^2+p.^2); %displacement vector! and its 1st derivative Z3=[R*cos(pi/2-4*lambda+(4*PHI)) R*sin(pi/2-4*lambda+(4*PHI))]; dZ3=[-R*sin(pi/2-4*lambda+(4*PHI)) R*cos(pi/2-4*lambda+(4*PHI))]; Y1=dZ3(1:70)'; X1=dZ3(71:140)'; dZ4=sqrt(Y1.^2+X1.^2); %check if displacement vector is correct figure(26) clf hold on plot(x3,y3, 'r' ,Z3(:,1),Z3(:,2), 'b' ); d_Ctheta_Phi=(sec(Phi)).^2./((sin(Theta)./((cot(Gamma*lambda)+cos(Theta ).*tan(Gamma*lambda)))+(((1cos(Theta)).*(sin(Theta).*tan(Gamma*lambda)))./((cot(Gamma*lambda)+cos( Theta).*tan(Gamma*lambda)).^2)))); d_Phi_Ctheta=(sin(Theta)./((sec(Phi)).^2.*(cot(Gamma*lambda)+cos(Theta) .*tan(Gamma*lambda)))+(((1cos(Theta)).*(sin(Theta).*tan(Gamma*lambda)))./((sec(Phi)).^2.*(cot(Gam ma*lambda)+cos(Theta).*tan(Gamma*lambda)).^2))); %d_Phi_Ctheta2=(1/(sec(Phi)).^2)'.*((sin(Theta)./(cot(Gamma*lambda)+cos (Theta).*tan(Gamma*lambda)))+(((1cos(Theta)).*(sin(Theta).*tan(Gamma*lambda)))./((cot(Gamma*lambda)+cos( Theta).*tan(Gamma*lambda)).^2))); %Checking

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147 Appendix F: (continued) diff_T_P=diff(Theta(2:end))./diff(Phi(2:end)); %diff_T_P=(Theta(4:end)-Theta(2:end-2))./(Phi(4:end)-Phi(2:end-2)) diff_P_T=diff(Phi(2:end))./diff(Theta(2:end)); %diff_P_T=(Phi(4:end)-Phi(2:end-2))./(Theta(4:end)-Theta(2:end-2)) check1=diff_T_P-d_Ctheta_Phi(3:end); check2=diff_P_T-d_Phi_Ctheta(3:end); check3=(ones(69,1)./d_Ctheta_Phi(2:end))-d_Phi_Ctheta(2:end); check4=(180/pi*acos(dot(dZ3',F_r3')./(dot(dZ3',dZ3').*dot(F_r3',F_r3')) .^.5))'; %%Check deffirentiation of cap theta with respect to phi figure(27) clf hold on plot(diff_T_P, 'r' ) plot(d_Ctheta_Phi(3:end), 'b' ) %Check deffirentiation of phi with respect to cap theta figure(28) clf hold on plot(diff_P_T, 'r' ) plot(d_Phi_Ctheta(3:end), 'b' ) %WORK done by Ft dWf=dot(Ft_3',dZ3'); dWf2=dot(F3',dZ3'); check5=(dWf-dWf2)'; %CALCULATE K T=(((dWf)'+momz3).*d_Phi_Ctheta); %%% captheta in embedded in the Torque!!! T1=(((dWf)').*d_Phi_Ctheta); T2=((momz3).*d_Phi_Ctheta); check6=T1+T2-T; %Cap theta vs components of T figure(29) clf hold on plot(Theta(2:end)*180/pi,T1(2:end), 'r' ) plot(Theta(2:end)*180/pi,T2(2:end), 'b' ) plot(Theta(2:end)*180/pi,momz3(2:end), 'g' ) %CALCULATE CONSTA Ktheta, KF, KM NONDIMENSIONALIZATION

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148 Appendix F: (continued) Ttheta=(T*R*4*lambda)/(E*I); TF=(T1*R*4*lambda)/(E*I); TM=(T2*R*4*lambda)/(E*I); check7=TF+TM-Ttheta; [CsKTHE,s]=POLYFIT(Theta(2:end),Ttheta(2:end),2); [CsKf,s1]=POLYFIT(Theta(2:end),TF(2:end),1); [CsKm,s2]=POLYFIT(Theta(2:end),TM(2:end),2); KTHEf = polyval(CsKTHE,Theta); Kff = polyval(CsKf,Theta); Kmf = polyval(CsKm,Theta); %shows tah he function of Ttheta, Tm and Tf fits the data. figure(30) clf hold on plot(Theta(2:end)*180/pi ,Ttheta(2:end), 'b' ) plot(Theta(2:end)*180/pi ,TF(2:end), 'r' ) plot(Theta(2:end)*180/pi ,TM(2:end), 'g' ) plot(Theta(2:end)*180/pi ,KTHEf(2:end), 'b*' ) plot(Theta(2:end)*180/pi ,Kff(2:end), 'r*' ) plot(Theta(2:end)*180/pi ,Kmf(2:end), 'g*' ) %%%shows the behavior of K and Ktheta, KF, KM figure(31) clf hold on plot(T(2:end), 'r' ) plot(T1(2:end), 'c' ) plot(T2(2:end), 'k' ) plot(Ttheta(2:end), 'b' ) plot(TF(2:end), 'm' ) plot(TM(2:end), 'g' ) %shows behavior of T and components wrt theta figure(32) clf hold on plot(Theta(2:end) ,Ttheta(2:end), 'b' ) plot(Theta(2:end) ,TF(2:end), 'r' ) plot(Theta(2:end) ,TM(2:end), 'g' ) plot(Theta(2:end) ,dWf(2:end), 'k' ) plot(Theta(2:end) ,momz3(2:end), 'c' ) %%%Finds componets fuctions of Ttheta Km=Kmf*E*I./(R*4*lambda) Kf=Kff*E*I./(R^2*4*lambda)

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149 Appendix F: (continued) %Find moment and force using componet function Km and Kf Mom=(Km.*d_Ctheta_Phi); Foc=(Kf.*d_Ctheta_Phi); %%%plots nondimenzionalized K componetss figure(33) clf hold on plot(TF(2:end), 'r' ) plot(TM(2:end), 'b' ) %%%plots function components of K theta figure(34) clf hold on plot(Kf(2:end), 'r' ) plot(Km(2:end), 'b' ) %plot moment from the data and the moment found using the model figure(35) clf hold on plot(Theta(2:end)*180/pi,Mom(2:end), 'r' ) plot(Theta(2:end)*180/pi,momz3(2:end), 'b' ) %plot(T2.*d_Ctheta_Phi,'k')%%%%ASK Dr. Lusk %plot force from the data and the force found using the model figure(36) clf hold on plot(Theta(2:end)*180/pi,Foc(2:end), 'r' ) plot(Theta(2:end)*180/pi,Ftan(2:end), 'b' ) %plot(T1/R.*d_Ctheta_Phi,'k') R h2 E b2 I lambda Cstheta Csgamma total_erroro CsKTHE CsKf CsKm %%%%TEST%%%%%%%%%%%

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150 Appendix F: (continued) Ctheta2= acos((tan(PHI).*cot(Gamma*lambda)+1)./(1+tan(PHI).*tan(Gamma*lambda))); %%%usi ng model or data? d_Ctheta_PHI=(sec(PHI)).^2./((sin(Ctheta2)./((cot(Gamma*lambda)+cos(Cth eta2).*tan(Gamma*lambda)))+(((1cos(Ctheta2)).*(sin(Ctheta2).*tan(Gamma*lambda)))./((cot(Gamma*lambda)+ cos(Ctheta2).*tan(Gamma*lambda)).^2)))); d_PHI_Ctheta2=(sin(Ctheta2)./((sec(PHI)).^2.*(cot(Gamma*lambda)+cos(Cth eta2).*tan(Gamma*lambda)))+(((1cos(Ctheta2)).*(sin(Ctheta2).*tan(Gamma*lambda)))./((sec(PHI)).^2.*(cot (Gamma*lambda)+cos(Ctheta2).*tan(Gamma*lambda)).^2))); CsKf2=9.1455.*Ctheta2-1.4521.*ones(70,1); CsKm2=-2.4524.*Ctheta2.^2+3.5246.*Ctheta2-0.0498.*ones(70,1); CsKTHE2=-0.3244.*Ctheta2.^2+8.9923*Ctheta2-0.1640.*ones(70,1); errorf=(CsKf2-Kff)./Kff errorm=(CsKm2-Kmf)./Kmf errorKT=(CsKTHE2-KTHEf)./KTHEf Kf2=CsKf2.*E*I./(R^2*4*lambda); Km2=CsKm2.*E*I./(R*4*lambda); errorf2=(Kf2-Kf)./Kf errorm2=(Km2-Km)./Km Foc2=Kf2.*d_Ctheta_PHI; Mom2=Km2.*d_Ctheta_PHI; Theterr=(Ctheta2-Theta)./Theta Ferr=(Foc2-Foc)./Foc Merr=(Mom2-Mom)./Mom figure(37) clf hold on plot(Foc(2:end), 'r' ) plot(Foc2(2:end), 'b' ) %%%%need a constant? figure(38) clf hold on plot(Mom (2:end), 'r' ) plot(Mom2(2:end), 'b' ) figure(39) clf hold on plot(Ctheta2,Mom2, 'r' ) plot(Ctheta2,Foc2, 'b' )

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151 Appendix F: (continued) figure(40) clf hold on plot(Theta(2:end),Mom(2:end), 'r' ) plot(Theta(2:end),Foc(2:end), 'b' ) plot(Theta(2:end),Ftan(2:end), 'g' ) plot(Theta(2:end),momz3(2:end), 'k' ) plot(Ctheta2(2:end),Mom2(2:end), 'r*' ) plot(Ctheta2(2:end),Foc2(2:end), 'b*' ) Zout=[R*cos(pi/2-2*lambda+(2*PHI)) R*sin(pi/2-2*lambda+(2*PHI)) R*sin(2*beta)] %%%use full mode devided by 2 ask!! dZout=[-R*sin(pi/2-2*lambda+(2*PHI)) R*cos(pi/2-2*lambda+(2*PHI)) R*cos(2*beta)]; %% %%%THE END Csgamma B CI rsqrd total_erroro Cstheta CsKTHE CsKf CsKm

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152 Appendix G: Tables of Summary of Result s for a Specific Horizontal Buckling EndLoaded Curved Beam Table G.1. Summary of results of curved beam with Constant 0.7781 b-2 0.8105 0.05847 b-1 -0.0077 0.02985 b0 -0.0005 0.00846 b1 -0.0046 0.06974 b2 0.0001 0.04877 Coefficient of determination R2 99.96% Total error 2.4977e-10% C 1.2386 T (*) 0.8410 +7.2766 -0.0739 fT (*) 8.7428 -0.6312 mT (*) -0.0492 +0.0746 -0.0048

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153 Appendix G: (continued) Table G.2. Summary of results of curved beam with Constant 0.7812 b-2 0.7441 0.1764 b-1 0.0235 0.0979 b0 0.0235 0.0312 b1 0.0634 0.1999 b2 -0.0278 0.1344 Coefficient of determination R2 99.92% Total error 1.4426e-8% C 1.2451 T (*) 0.7486 +7.4219 -0.0909 fT (*) 8.7603 -0.6694 mT (*) -0.1973 +0.2982 -0.0183 Table G.3. Summary of results of curved beam with Constant 0.7864 b-2 0.4210 0.3802 b-1 0.1709 0.2267 b0 -0.0283 0.0794 b1 0.3792 0.4098 b2 -0.1446 0.2647 Coefficient of determination R2 99.54% Total error 1.4710e-7% C 1.2559 T (*) 0.6064 +7.6418 -0.1110 fT (*) 8.7916 -0.7361 mT (*) -0.4459 +0.6701 -0.0378

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154 Appendix G: (continued) Table G.4. Summary of results of curved beam with Constant 0.7938 b-2 -0.8429 0.7203 b-1 0.7667 0.4555 b0 -0.1280 0.1718 b1 1.5518 0.7422 b2 -0.5493 0.4614 Coefficient of determination R2 98.89% Total error 7.3009e-7% C 1.2711 T (*) 0.4304 +7.9087 -0.1287 fT (*) 8.8415 -0.8413 mT (*) -0.7980 +1.1904 -0.0600 Table G.5. Summary of results of curved beam with Constant 0.8033 b-2 -5.6781 1.2942 b-1 3.1644 0.8605 b0 -0.5591 0.3448 b1 5.8103 1.2796 b2 -1.9389 0.7671 Coefficient of determination R2 98.6% Total error 2.4244e-6% C 1.2908 T (*) 0.1990 +8.2504 -0.1502 fT (*) 8.9062 -0.9835 mT (*) -1.2459 +1.8406 -0.0743

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155 Appendix G: (continued) Table G.6. Summary of results of curved beam with Constant 0.8148 b-2 -26.4491 2.2828 b-1 14.0351 1.5874 b0 -2.6508 0.6697 b1 23.2103 2.1709 b2 -7.3468 1.2565 Coefficient of determination R2 98.93% Total error 6.3645e-6% C 1.3147 T (*) -0.0520 +8.6129 -0.1615 fT (*) 9.0057 -1.1829 mT (*) -1.7981 +2.6234 -0.0747 Table G.7. Summary of results of curved beam with Constant 0.8281 b-2 -142.1823 4.1447 b-1 78.0177 3.0046 b0 -15.7772 1.3276 b1 115.4554 3.7958 b2 -34.7151 2.1215 Coefficient of determination R2 99.44% Total error 1.5795e-5% C 1.3427 T (*) -0.3244 +8.9923 -0.1640 fT (*) 9.1455 -1.4521 mT (*) -2.4524 +3.5246 -0.0498

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