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A pseudo-rigid-body model for spherical mechanisms

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Title:
A pseudo-rigid-body model for spherical mechanisms the kinematics and elasticity of a curved compliant beam
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Book
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English
Creator:
Le‚àö‚â•n, Alejandro
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University of South Florida
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Tampa, Fla.
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Subjects / Keywords:
Compliant mechanisms
Large deflection
Virtual work
Mems
Out of plane
Dissertations, Academic -- Mechanical Engineering -- Masters -- USF   ( lcsh )
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bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

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Abstract:
ABSTRACT: This thesis improves a previous kinematic analysis and develops the elastic portion of the analysis of a curved compliant beam. This analysis is used to develop a Pseudo-Rigid-Body Model for the curved compliant beam. The Pseudo-Rigid-Body Model consist of kinematic and elastic parameters which can be used to simplify the computation of the large deflections of the beam as it undergoes spherical motion. The kinematic parameters that are developed are the characteristic radius, Gamma*length, the parametric angle coefficient, c_theta, and the kinematic parametrization limit, Capital_theta_max(Gamma). The elastic parameters developed are the stiffness coefficient, K_theta, and the elastic parameterization limit, Capital_theta_max(K). Additionally, curve fit parameters are developed which enable the calculation of the stress in curved beam as it deflects.
Thesis:
Thesis (M.S.)--University of South Florida, 2007.
Bibliography:
Includes bibliographical references.
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Mode of access: World Wide Web.
Statement of Responsibility:
by Alejandro Le‚àö‚â•n.
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Title from PDF of title page.
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Document formatted into pages; contains 82 pages.

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aleph - 001935515
oclc - 226250593
usfldc doi - E14-SFE0002272
usfldc handle - e14.2272
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ABSTRACT: This thesis improves a previous kinematic analysis and develops the elastic portion of the analysis of a curved compliant beam. This analysis is used to develop a Pseudo-Rigid-Body Model for the curved compliant beam. The Pseudo-Rigid-Body Model consist of kinematic and elastic parameters which can be used to simplify the computation of the large deflections of the beam as it undergoes spherical motion. The kinematic parameters that are developed are the characteristic radius, Gamma*length, the parametric angle coefficient, c_theta, and the kinematic parametrization limit, Capital_theta_max(Gamma). The elastic parameters developed are the stiffness coefficient, K_theta, and the elastic parameterization limit, Capital_theta_max(K). Additionally, curve fit parameters are developed which enable the calculation of the stress in curved beam as it deflects.
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A Pseudo Rigid Body Model For Spherical Mechanisms: The Kinematics and Elasticity of a Curved Compliant Beam by Alejandro Le n A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Depa rtment of Mechanical Engineering College of Engineering University of South Florida Major Professor: Craig P. Lusk Ph.D. Autar K. Kaw, Ph.D. Nathan Crane, Ph.D. Date of Approval: November 6, 2007 Keywords: complia nt mechanisms, large deflection, virtual work mems out of plane Copyright 2007 Alejandro Len

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Dedication To my wife, Kathryn

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Acknowledgements I wish to express my gratitude to everyone who contributed to making this thesis a reality. First, my advisor Dr. Craig P. Lusk who patien tly guided me and took me under his wing from the beginning to bring this thesis to fruition. He is a true role model. I also want to thank my committee members Dr. Nathan Crane and Dr Autar K. Kaw for taking the time to read and provide me with feed back to improve this thesis. I would like to thank my brother, Hector Le n for all of his guidance and advice through out my life you have made me a better man and have indirectly prepared me to tackle challenges like this thesis. Moreover, I would like to thank my parents, whom have loved me and supported me all my life, I love you guys so much, thank you. Additionally, I would like to thank my friend and lab mate, Joseph Choueifati you have been essential in keeping me m otivated and without your shoulder to lean on, this thesis would have been much more difficult, thank you. Finally, I would like to thank the Mechanical Engineering Department., the College of Engineering and University of South Florida, Tampa Florida

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i Table of C ontents List of Tables ................................ ................................ ................................ .................... ii i List of Figures ................................ ................................ ................................ .................... iv Abstract ................................ ................................ ................................ .............................. v i Chapter 1. Introduction ................................ ................................ ................................ ........ 1 1.1 Objective ................................ ................................ ................................ ............ 1 1.2 Motivation ................................ ................................ ................................ .......... 2 1.3 Scope ................................ ................................ ................................ .................. 2 1.4 Contributions ................................ ................................ ................................ ...... 3 1.5 Roadmap ................................ ................................ ................................ ............ 3 Chapter 2. Background ................................ ................................ ................................ ........ 5 2.1 Pseudo Rigid Body ................................ ................................ ............................ 5 2.2 Kinematic portion for a Ps eudo Rigid Body M odel ................................ .......... 6 2.3 Elastic p ortion for a Pseudo Rigid Body M ode l ................................ ............... 8 Chapter 3. Methodology and Model Development ................................ .............................. 9 3.1 Correspondence between spherical and planar PRBMs ................................ .... 9 3.2 Model description of kinematics of compliant curved beam ........................... 10 3.3 Sp herical Pseudo Rigid Body Model kinematics ................................ ............ 13 3.4 Finite Element Model ................................ ................................ ...................... 15 3.5 Pseudo Rigid Body Model parameters ................................ ............................ 17 Chapter 4 Elastic Portion of a PRBM for a Spherical Mechanism ................................ ... 22 4.1 Introduction ................................ ................................ ................................ ...... 22 4.2 Principle of virtual work ................................ ................................ .................. 22 4.3 Stiffness C oefficient for a Spherical Pseudo Rigid Body Model .................... 26 Chapter 5. S pherical PRBM Stress A nalysis ................................ ................................ ..... 29 5.1 Introduction ................................ ................................ ................................ ...... 29 5.2 Development of bending stress for a spheric al PRBM ................................ .... 30 5.3 Axial stress development for a spherical PRBM ................................ ............. 32 5.4 Displacement cons traint ................................ ................................ ................... 33 5.5 Maximum Stress A nalysis ................................ ................................ ............... 35 Chapter 6. Results and Discussion ................................ ................................ ..................... 3 7 6.1 Kinematic improvements ................................ ................................ ................. 37

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ii 6.2 Elastic portion ................................ ................................ ................................ .. 39 6.3 Stress analysis ................................ ................................ ................................ .. 40 6.4 Conclusion ................................ ................................ ................................ ....... 41 References ................................ ................................ ................................ .......................... 4 2 Appendices ................................ ................................ ................................ ......................... 45 App endix A: DETC paper ................................ ................................ ..................... 46 Appendix B: ANSYS batch code ................................ ................................ ........... 57 Appendix C: MATLAB code ................................ ................................ ................. 64 for spherical triangles ................................ ................. 77

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iii List of Tables Table 1 Position and ori entation of coordinate frames ................................ ........... 12

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iv List of Figures Figure 2.1 A PRB M link for a cantilever beam ................................ ............................ 6 Figure 2 .1a Cantilever segment with vertical force at the free end ................................ 7 Figure 2.1b The PRBM equivalent ................................ ................................ .................. 7 F igure 3.1 Reference frames describing the motion of the end of a compliant circular cantilever beam ................................ ............................ 11 Fi gure 3.2 Sp herical mechanism with its respective kinematic definitions ................ 13 Figure 3.3 The Pseudo Rigid Body M odel of the compliant curved beam ................ 15 Figure 3.4 Determination of error in the deflection approximation ............................ 18 Figure 3.5 The characteristic radius factor , versus the arc angle of the compliant beam ................................ ................................ ......... 20 Figure 3.6 The parametric angle coefficient ,versus the arc angle of the compliant beam ................................ ................................ .......... 21 Figure 3.7 The paramet erization limit as a function of the arc angle subtended by the circular compliant beam, ................................ 21 Figure 4.1 Spherical PRBM equivalent ................................ ................................ ....... 24 Figure 4.2 versus Lambd a ................................ ................................ ............ 28 Figure 5.1 Three dimensio nal stress element at point P ................................ ............. 29 Figure 5.2 v ersus for variable and aspect ration 0.1 ................................ ... 34 Figure 5.3 versus for variable and aspect ration 0.4 ................................ ... 34 Figure 5.4 versus for variable and aspect ration 0.7 ................................ ... 35

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v Figure 6.1 The characteristic radius factor ,versus the arc angle of the compliant beam ................................ ................................ 37 Figure 6.2 The parametric angle coefficient, ,versus the arc angle of the compliant beam, ................................ ............................... 38 Figure 6.3 The parameterization limit as a function of the arc angle subtended by the circular compliant beam, ................................ 38 Figure 6.4 versus Lambda ................................ ................................ ........... 39 Figure 6.5 versus Lam bda ................................ ................................ ............ 40

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vi A Pseudo Rigid Body Model For Spherical Mechanisms: The Kinematics and Elasticity of a Curved Compliant Beam Alejandro Le n ABSTRACT This thesis improves a previou s kinematic analysis and develops the elastic portion of the an alysis of a curved compliant beam. This analysis is used to develop a Pseudo Rigid Body Model for the curved compliant beam. The Pseudo Rigid Body Model consist of kinematic and elastic parameters which can be used to simplify the computation of the larg e deflections of the beam as it undergoes spherical motion. The kinematic parameters that are developed are the characteristic radius, the parametric angle coefficient, ,and the kinematic parametrization limi t, The elastic parameters developed are the stiffness coefficient, and the elastic parameterization limit, Additionally, curve fit parameters are developed which enable the calcula tion of the stres s in curved beam as it deflects

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1 Chapter 1 Introduction be useful, but they ] I n 1959, physicist Richard Feynman challenged re searchers to explore the possi bilities of technological advancement on the small scale. Since that time computers have been reduced from room size to desktop and hand held devices, facilitated by ever sh rinking microelectronics. Micro Electromechanical N micro mirrors, each acting as a single pixel), which is used in some computer projectors. Other example s are pressure sensors and a ccelerometers which ar e used in applicat ions such as automotive airbags [5]. The integration of compliant mechanisms into MEMS fabrication stands to make a contribution in the design and performance of MEMS 1.1 Objective The objective of this thesis w as to develop the elastic parameters for a spherical Pseudo rigid bod y M odel to expedite and simplify the analysis and design of compliant mechanisms In addition, to develop the supporting software codes for finite element analysis (FEA) and the analysis of the FEA data.

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2 1.2 Motivation Most mechanisms are systems of levers cams and gears, which move, rotate, and have rigid parts. On the other hand, c ompliant mechanisms some or all of their ability to move from the deflection of flexible segments Compliant mechanisms m a y have improved performance and lower cost when compared with rigid body mechanisms [2, 3]. Hence, the ability to expedite and simplify the analysis and design of compliant mechanisms could prove benefici al to designers. An area where the benefits could be greatly implemented would be in MEMS fabrication. MEMS can benefit a great deal in integrating compliant mechanisms in their designs. The use of compliant mechanisms allows MEMS to be easily fabricate d, eliminates the use of hinges, allows for precise motion control, and practically eliminates wear [7]. 1.3 Scope Compliant mechanism theory permits a procedure called rigid body replacement, in which two or more rigid links of the mechanism are replaced by a compliant flexur e with equivalent motion [7] Methods for designing flexure s with equivalent motion to replace rigid links are detailed in Pseudo Ri gid Body Models (PRBMs). In several texts, [16,22] rigid body analysis and synthesis techniques have been cla ssified as planar, spherical or spatial according to the type of vector algebra used to describe the mechanisms. In a planar mechanism, the path of any single part of a link lies in a plane and in a spherical mechanism, the path of any single part of a link lies on the surface of a sphere [17]. Numerous PRBMs have been developed for planar mechanisms [1, 9, 12, 13, 20] and used to design compliant mechanisms in applications such as prosthetics [25], clutches [26, 27], micro bearings [28], and bist able mechanisms [29]. Thus, extensive

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3 research has been done on planar compliant mechanisms using planar PRBMs. For MEMS applications that require mechanisms which rotate out of plane of fabrication such as the with an in plane ro tational input, or that rotate spatially about a point spherical mechanisms may prove appropriate [18]. It is believe d that the description of PRBMs for spherical compliant mechanisms will facilitate the design of MEMS with out of plane motion. In this thesis, a PRBM for a spherical compliant mechanism is developed. The kinematics of a curved flexure with the equi valent of a vert ical end load have been described and the elastic parameters of the PRBM have been developed. 1.4 Contributions This thesis h as four major contributions to the analysis and design of compliant mechanisms. First, i mprovements to the PRBM kinematics developed by Jagirdar [17] were made. Where a robust form for calculating the y axis rotational displacement, was developed, thus permitting a calculation of the horizontal angle Second, the development of the elastic parameters for the PRBM of a spherical compliant mechanism. Thirdly, a FEA analysis was preformed to validate the PRBM. A s a result an ANSYS batch code had to be developed in order to allow large load steps runs for each aspect ratio of the compliant beam. Finally, a MATLAB code had to be developed in order to analyze the data produced by the FEA analysis. 1.5 Roadmap T his chapter has served as a general introduction to the work done in this thesis. Chapter 2 will introduce the background for PRBMs and the spherical kinematics. Chapter 3 will describe the analogy between a planar PRBM and a spherical PRBM and

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4 detail th e improvements made to Jagirdar [17]. Also, the FEA nomenclature, constraints used in the PRBM and its results will be presented. Chapter 4 will discuss the principle of virtual work and how it was coupled with the PRBM to develop the stiffness coefficie nt, In addition, chapter 4 will discuss the derivation of the governing equations for the FEA analysis of Chapter 5 describes the stress calculation for the spherical compliant beam and the derivations of the rotation matrices needed for the stress calculations based on the FEA data. Finally, chapter 6 will discuss the conclusions based on the results.

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5 Chapter 2 Background 2.1 Pseudo Rigid B ody Closed form solutions for large deflections in cantilever beams have been developed in the past in the form of ellipti c integrals. While elliptic integrals offer closed form solutions to large deflection problems they are limited to relative simple geometries and loadings a nd thei r derivations are laborious I t was observed that in all large deflections of cantilever beams the path of the beam end is nearly circular, with a center of curvature at some point o n the undeflected part of the beam. This observation serves as the catalyst that leads to the developme n t of the pseudo rigid body model (PRBM) which allows for the motion of the end of the cantilever beam to be accurately approximated. The PRBM consists of diagr ams and equations describing the motion and stiffness of a compliant member in terms of a rigid l ink equivalent mechanism which has the same motion and stiffness for a known range of motion and to a known mathematical tolerance. A PRBM can be used to perform analysis (i.e. given a compliant flexure, its motion can be found by treating it as the rigid body) or design (given a particular desired motion, a rigid body mechanism that performs the motion can be found, and the PRBM can be used to convert that rigid body mechanism into a compliant mechanism). Once the rigid body analogue to a compliant segmen t has been identified, the kinematic and

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6 elastic parameters of the PRBM are optimized and validated so that range of applicability and level of error is known and acceptable [17] 2.2 Kinematic p ortion for a Pseudo Rigid Body M odel The PRBM al lows a simplification of the bending of a compliant beam, by having an equivalent rigid body link rotate about a characteristic pivot located a distance away from the free end of the beam w here is the charac teristic radius and is the characteristic radius factor [17] The Pseudo Rigid Body Model includes the development of a pseudo rigid body angle, which is the amount of rotation about the pseudo pivot as shown in Figure 2.1 Figure 2.1 A PRBM link for a cantilever beam The creation of such PRBMs is justified because they are easy to use in design and the use of the PRBM in connection with rigid body synthesis techniques produces compliant mechan ism configurations that are unlikely to be produced in any other way [17]. An example of a PRBM f or a straight cantilever beam with vertical end load [7] is shown in Figure 2.2 Figure 2.2 (a) shows a straight cantilever beam subjected to a vertical end load F Figure 2.2 (b) shows the pseudo rigid body equivalent of the straight cantilever

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7 beam. The distance from the fixed end to th e free end in the x direction is a the distanc e from the fixed end to the free end in the y direction is b length of the st raight beam is l is the pseudo rigid body angle and is the characteristic radius factor. The angle of inclination of the beam at the beam end is given by Figure 2.1a Cantilever segment with ver tical Fig ure 2.1b T he PRBM equivalent force at the free end. The coordinates of the beam end of the compliant beam are given in terms of the PRB M angle, as: a = l [1 (1 cos )] (2.1) b = l sin (2.2) Where =0.85 for a vertical end load. The relationship between and is given by: = 1.24 ( 2 .3) These relations are accurate to less than 0.5% error for 7] describes the criteria for calculating an acceptable value for the characteristic radius factor, by first determining the maximum acceptable percentage error in deflection. The value for that would allow the maximum pseudo rigid body

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8 angle, whi le still keeping the error in the position of the beam end less than 0.5% can be calculated by: (2.4) As stated by [7, 9] and subject to the parametric constraint for 0< < (2.5) where is the relative deflection error, and is defined as the vector difference of the deflected position of point P and its original undeflected position. 2.3 Elas tic portion for a P seudo Rigid Body M odel The elastic portion of the PRBM includes a nondimensionalized torsional spring constant , called the stiffness coefficient as shown on Figure 2.3 Combined with geometric and material properties, the stiffness pseudo rigid body model. facilitates the calculation of the force required t o deflect the rigid body system that is equivalent to the force needed to deflect the compliant beam [18 ] The elastic portion also yields a for accurate force prediction. An implicit advantage of the parametric approximation of the pseudo rigid body model is that it can be used to obtain accurate initial estimates hence circumventing the laborious method of elliptic integrals.

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9 Chapter 3 Methodology and Model Development 3 .1 Correspondence between spherical and planar PRBMs In p lanar mechanisms the joint axes are parallel ; in spherical mechanisms the joint axes intersect at the center of the sphere [16] In planar compliant mechanisms, this characteristic is usually achieved by designing straight flexures that, at each point along their length are most f lexible about parallel lines and are considerably more rigid in other directions. In s pherical compliant mechanisms, this characteristic can be achieved by designing curved flexures that, at each point along the arc, are most flexi ble about lines that point to the centre of the sphere. In both kinds of mechanisms it is necessary that the length (arc length) of the flexure be much greater than the width of the beam and the width of the beam to be larger than its thickness [17] It is hypothesized that a flexure which is a long, thin circular arc will move in a manner consistent with spherical kinematics when appropriately loaded [17] There is a correspondence principle between spherical PRBMs and planar PRBMs. The correspondence p rinciple is that when small angle assumption is used for spherical arcs. i.e. the arc length is much smaller than the radius of the sphere, the spherical PRBM becomes identical to planar PRBM. To emphasize the relationship between lines and arcs in this t hesis the lengths in planar model are denoted with Roman letters, and the equivalent arcs in the spherical model are denoted wit h the Greek letter equivalents. For example the arc length, that appears in some formulas for spherical

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10 mechanisms, can be related to the planar length, b Thus, using small angle approximation: (3 .1) (3 .2) Where b is the planar equivalent of the a rc Similarly a and l are the planar equivalent of arcs and respectively. Additionally, similar terminology is used in planar and spherical PRBMs, for angles between lines (arcs) such as , and for ratios such as and These variables do not change in the small angle case. In the planar case, the deflected angle of beam end, is about an axis normal to the plane. Similarly, in the spherical case, the deflection of the beam end, is about an axis normal to the tangent plane to the sphere at the beam end. 3 .2 Model description of kinematics of compliant curved beam In order to implement the PRBM some reference frames must first be identified. The following nomenclature was developed by Saurabh Jagirdar [17] for the implementation of the PRBM in the kinematic analysis of a compliant curved beam, and will further be used in this thesis for the kinetic analysis. T he kinematics of the compliant circular cantil ever, PQ is de scribed by using a series of co ordinate frames, as shown in Figure 3.1

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11 Figure 3.1 Reference frames describing the motion of the end of a compliant circular cantilever beam The fixed end of the curved cantilever beam is deno ted as P and free end of the beam as Q Let the center of a sphere be defined by frame O and the frames A B C and D are on the surface of the sphere. The pos ition and orientation of the co ordinate frames are related as follows [18] :

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12 Table 1 Position and orientation of coordinate frames Frame Frame Description s [17] O This frame is attached to the center of the sphere. The axis passes through the un deflected beam end Q. The axis is normal to the plane containing the un deflected beam. The axis completes the right handed orthogonal triad and is parallel with the neutral axis of the un deflected beam end at Q. A This frame has the same orientation as th e O frame and is located at the end of the un deflected beam. B This frame locates the deflected position of the beam end Q in the a plane (analogous to the translation in the x direction in the planar model C This frame describes movement of beam end Q in the plane rotating about point O (analogous to the translation in the y direction in the planar model) D This frame is at the same position as the C frame and tracks the rotation of the beam end about the radial axis through the beam end (analogous to the rotation of the beam end about the z axis in the planar model) E This frame describes relative position of beam end P with respect to the other frames T he frames are described by the matrices A B C and D where the columns of the matrix are the basis vectors. The transformations relating the frames are given by: (3 .3) (3 .4) (3 .5) (3 .6)

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13 Thus the motion of the curved cantilever beam is described by the parameter s and which are analogous to planar parameters l a b and respecti vely These parameters are shown in Figure 3.1 Moreover, a n additional frame is needed for the elastic portion of the PRBM. Frame E which is located at a point P, typically is the point of highest bending stress. (3.7) 3 .3 Spherical Pseudo Rigid Body Model kinematics An example of the implementation of the PRBM f o r a spherical compliant mechanism can be seen in Figur e 3.2 with its respective kinematic definitions. Figure 3.2 Spherical mechanism with its respective kinematic definitions

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14 The angle is defined as the pseudo rigid body angle for both the planar and the spherical PRBMs. The characteristic radius factor, is defined as the length from the beam end to the pseudo pivot divided by the length of the beam neutral axis subtends an arc, and its PRBM consists o f two rigid links, which make up circular arcs of and and are joined by a revolute joint which is located at center O [18]. The relationships for and in terms of and right spherical triangles [19] on the triangle shown in Figure 3.2 with sides and One finds the y axis rotational displacement, as a function of and as: (3 .7) The horizontal angle : (3 .8) The vertical angle : (3 .9) Equations 3.7 and 3 .8 are the result of improvements made to the PRBM kinematics for calculating first deve loped by Jagirdar [17]. Equation 3 .7 is a robust form for calculating which removes singularities at Equation 3 .8 then allows for the calculation of from

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15 3 .4 Finite Element Model To compute the deflection of the curved beam undergoing spherical motion, the beam was modeled in a commercial FEA software package (ANSYS 11). Three dimensional beam elements with six degrees of freedom (BEAM 4) were used. Becau se large deflection analysis was required, the iterative nonlinear solver was used. The loading conditions proved easiest to apply using the double slider model. A load was applied to the FEA model such that there is no reaction load at the fixed end, P and the free end, Q of curved cantilever beam moves in a manner consistent with spherical kinematics. T he load direction depends on the d isplacement of the beam end, and the displacement of the beam end depends on the load direction. Thus, to ensure tha t there is no reaction load at the fixed end, P we need to know the path followed by the beam end as shown in Figure 3.3 Figure 3.3 The Pseudo Rigid Body M odel of the compliant curved beam

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16 The path followed is an arc on the sphere from the A f rame (undeflected position Q ) to the C or D frame (final position of Q ). When the beam PQ is taken as fixed at P the A frame of reference is fixed. The motion of the beam can also be described in the B frame of reference such that the end Q of the beam i s allowed to move in the plane only Also, point Q was constrained to rotate an amount about the O 3 axis and was permitted to rotate freely about the O 1 and O 2 axes. Thus, t he input parameter is the angle which determines both the rotation of the beam end Q and its deflection. One output obtained is the rotation angle, of the beam end P as it moves along a circular arc in the plane (pur e rotation about O 2 axis). The other output is the rotation, of the neutral axis of the beam at Q about the radial axis. T hat is, the beam undergoes spherical motion such that the ends P and Q move on orthogonal great circles. The simulation was repeated varying: a) The initial beam angle b) The aspect ration of the curved flexible beam (thickness divided by width ) Simulations were run for beam angles ranging from 4 degrees to 112 degrees in increments of one degree and aspect ratios of 0.1, 0.4, 0.7. The values of the input displacement, were applied in 1600 even increments ranging from 0 degrees up to the beam angle Additionally, in order t o efficiently perform the simulation an ANSYS batch code was developed. The batch code allows the user to automatically cycle through all the different angle s of and its corresponding aspect ratio. Otherwise, an user would be force to manually change and the aspect ratio for each run from 4 to 112 degrees. The parametric angle co efficient, the characteristic radius factor, and the

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17 parameterization limit, are obtained from the results of the FEA model using the procedure described in the next section. Additionally, the ANSYS code also calculated the force f c2 and the moment M for the elastic portio n discussed in chapter 4. 3 .5 Pseudo Rigid Body Model parameters The development of the spherical PRBM kinematics lends it self to develop the method [7] An optimal value for the characteristic radius f actor , may be found given a maximum acceptable percentage error in deflection. Based on equations 3.7 and 3.8 w e found the value of that would allow the maximum pseudo rigid body angle, while keeping the error in the position of the beam end less than 0.5 %. express the dependence of the pseudo rigid body angle, on for the spherical PRBM. From [18] we have: (3.10) (3.11) Therefore, (3.12) which is subject to the parametric constraint (3.13)

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18 where is the relative deflection error, and is defined as the vector difference of the deflected position of point Q and its original undeflected position. The deflection obtained using FEA, is shown in Figure 3.4 and can be calculated from the beam m otion parameters, and [18] as: (3.14) Figure 3.4 Determination of error in the deflection approximation The PRBM deflection estimate, is given by the vector difference of the deflected location of point Q calculated using the PRBM and its undeflected coordinates, or T he position of point Q according to the PRBM is where R is the

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19 rota tion operator that acts on the vector and rotates it about the pseudo pivot, through an angle [18]. The action of the operator R on is given [30] by: (3.15) where and which reduces to (3.16) Therefore [18], (3.17) The error is defined as magnitude of the vector difference between the final positions of the curved flexible segment found using FEA and the final position found using the Pseudo Rigid Body Model with a particular value of Thus, the error in the deflection is calculated as: (3.18) The relative error is defined as: (3.19) Appendix C gives the MATLAB code for finding the parameterization limit accuracy the value of at which the error in the model approximation exceeded 0.5%.

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20 Furthermore, based on equations 3.7 and 3.8, refinements to the FEA code, and the use of improvements in the calculations of parameters , and were made. Figures 3.4, 3.5, and 3.6 illustrate the improved calculations for , and respectively, as well as for each aspect ratio. Figure 3.5 The characteristic radius factor, versus the arc angle of the compliant beam

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21 Figure 3.6 The parametric angle coefficient, versus the arc angle of the compliant beam, Figure 3.7 The parameterization limit as a function of the arc angle subtended by the circular compliant beam,

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22 Chapter 4 Elastic Portion of a PRBM for a Spherical Mechanism 4.1 Introduction This thesis uses the principle of virtual work and the Pseudo Rigid Body Model concept to develop force deflection relationships for compliant mechanisms. Why do we use virtual work, as oppose to the well known Newtonian approa ch? It is worth mentioning, that one of the main advantages of using the virtual work approach is that the system, in this case the spherical compliant mechanism, is viewed and analyzed from an energy perspective and thus the end reaction forces are not n ecessary because they do not move and hence do no work, greatly simplifies the force deflection analysis. 4.2 Principle of virtual work work of all active forces is zero if a nd only if an ideal mechanical system is in n t s do no work where it can be expressed as The compliant mechanism in this thes is will be assumed to be ideal. In order to evaluate virtual work, a virtual displacement must first be consider ed A virtual displacement is a small, arbitrary displacement, which is expressed as a

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23 function of the generaliz ed coordinates. The virtual work, due to a n applied force and a virtual displacement, is: (4.1) Similarly, the virtual work due to a n applied moment, and a virtual angular displacement, is: (4.2) A force is conservative if the work done by the force is independent of path, that is, dependent on the co ordinate of the displacement endpoints only [7]. In this case, the work done is the difference in the potential energy, V of the system at the two endpoints: The work done on a spring fits this category. The strain energy of a sp ring may be determined from (4.3) where is the spring force as a function of s and is the value of s for which the spring force is zero. Torsional springs are also common in pseudo rigid body models, and their strain energy may be calculated in general form as: (4.4) where is the spring torque as a function of and is the value of for which the spring torque is zero. The PRBM in this thesis uses a l inear spring. The general nonlinear forms of equations (4.3) and (4.4) are [7]:

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24 (4.5) (4.6) This form is useful because integration and differentiation of the nonlinear spring functions are avoided. It is importan t to note that with a virtual work approach the number of generalized coordinate must be equal to the number of degrees of freedom of the system. The number of equations is also equal to the number of degrees of freedom [7]. The force displacement charac terist ics of the compliant mechanism i n this thesis were found by applying the principle of vi rtual work. The following derivations are provided in the context of the Pseudo Rigid Body Model of the compliant mechanism in Figure 4.1. Figure 4.1 Sp herical PRBM equivalent A force, f c 2 was applied parallel to the at the end point Q, an output moment, M about and a linear torsional spring to represent a small length flexural pivot at with a

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25 spring constant of In this thesis the force f c2 and the moment M were calculated using an ANSYS batch code as described in chapter 3 (see Appendix B ) In order to apply t he principle of virtual work, the input rotati on, was chosen as the generalized coordinate. Next we write the force f c2 and M in vector form as: (4.7) (4.8) The virtual displacement, is found by writing a vector , from a fixed point, the origin to the p lacement of each force, as follow s : (4.9 ) Next the virtual displacement was found by differentiating the position vector with respect to the generalized coordinate (4. 10 ) T he virtual work due to the force f c2 was calculated by taking the dot product of the force vector and the vi rtual displacement. The virtual work due to the moment was added to the virtual work done by the applied force f c2 Equation 4.11 is the result. (4.11 ) Until this point we have acc ounted for the virtual work done by the applied forces and the moments, but we have yet to take into account the virtual work done by the spring. In the next section we will look at the virtual work done by the spring and how it is derived.

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26 4 3 Stiffne ss Coefficient for a S pherical Pseudo Rigid Body Model constant, called the stiffness coefficient Combined with geometric and material properties, the stiffness coefficient is used to determine the value of the spring constant K, Pseudo Rigid Body Model [7]. To calculate the principle of virtual work in conjunction with the PRBM concept is used to develop force deflection relationships for compliant mechanisms as outlined in [20]. Moreover, the PRBM in this thesis includes a linear spring to model the stiffness of the beam and quantify its opposition to f c2 From equation 4.6 and as shown in [7] and letting = s virtual work equals: (4.12) where is as follows: For (4.12a) taking the derivative of equation 4.12a yields: ( 4.12b) solving for and taking the reciprocal: (4.12c)

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27 Thus, the total virtual work of the system is the sum of all the virtual work components: (4.13) where and substituting into equation 4.13 we can then solve for the parameter of interest. In this case (4.14) w here is a nondimensionalization factor. We further separate into its contributing components (the stiffness contributed by the moment) and (the stiffn ess contributed by the force f 3 ) and graph them with respect to where (4.15) (4.16) as see n in Figure 4.2

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28 Figure 4. 2 versus Lambda

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29 Chapter 5 Spherical PRBM Stress A nalysis 5.1 Introduction One of the most critical aspects of compliant mechanism design is ensuring that the mechanism will undergo its specified task without failing [7]. To this end, the stress analys is for the spherical PRBM has been developed. In any mechanism the area most likely to fail is the area that experiences maximum stress. For the curved compliant beam point P is the point of maximum stress. Figure 5.1 shows a three dimensional stress e lement w ith its axes aligned with side, top and frontal axis of the beam, respectively at point P. The vertical shear forces, and shown on F igure 5.1 make a negligible contribution t o the overall stress at point P, thus will not be included in the stress analysis. Figure 5.1 Three dimen sional stress element at point P

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30 5 .2 D evelopment of bending stress for a spherical PRBM As shown on Figure 5.1 p oint P experiences s everal different forces and moments that cause stress; in this section the stress caused by moments and are calculated A n additional frame, E, was developed in order to locate and des cribe the movement of point P, as shown in equation 3.7. Restated for reference: (5.1) In order to calculate the moment at P caused by a force at frame C, several frame transformations took place. First, t h e position vector locates point Q with respect to point P and was developed as follows : (5.2) where substituting for the unit vectors given by equation 3.5, wi th respect to frame C in to equation 5.2 yields: (5.3) equation 5.3 simplifies to: (5.4) then to describe the moment at P caused by a force at C with respect to frame C we take where is calculated from the ANSYS code and will be taken in the general f orm of

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31 (5.5) thus the yields an expression of the moment at P caused by a force at C, with respect to frame C as follows: (5.6) Similarly, a mo ment at P caused by a force at C with respect to frame B : (5.7) multiplying t hrough and rearranging equation 5.7 yields: writing in a more compact form yields equation 5.8, (5.8) Finally, the moment at P caused by a force at C with respect to frame E where is the rotation about O described by equation 5.1, thus we have: (5.9)

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32 equation 5.9 simplifies to: (5.10) equation 5.10 shows the individu al contribution of and to the overall bending stress, hence: (5.11) (5.12) (5.13) where equation 5.13 can be seen as the torsion about the axis. 5.3 Axial stress development for a spherical PRBM The development for the axial stress is very similar to how the bending stress was d erived in the previous section. As shown in Figure 5 .1, point P experiences a load in the direction which creates an axial stress on the e 2 e 1 plane. The effects of the force at frame C on poin t P with respect to frame E were deri ved in a similar way as the bending stress. Starting with equation 5.5, where and substituting for vectors and yields: (5.14) repeating the substitution process for vectors and using equation 5.1 yields:

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33 (5.15) equation 5.15 shows the individual contribution of and to the overall axial stress, hence: (5.16) (5.17) (5 .18) 5.4 Displacement constraint An important parameter that emerges from the analysis of this spherical compliant mechanism is that a force is necessary for the mechanisms to remain in spherical motion. This is not a working forc taking into account for the elastic parameter calculations, but it contributes to the stress at point P. Force was plotted for several different beam angles and the parametric angle to better study the stress in the beam and its spherical motion. Figures 5.3, 5.4, and 5.5 are the graphs of for aspect ratio 0.1, 0.4, and 0.7 respectively.

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34 Figure 5.2 v ersu s for variable and aspect ratio 0.1 Figure 5.3 versus for variable and aspect ratio 0.4

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35 Figure 5.4 v ersu s for variable and aspect ratio 0.7 5. 5 Maximum S t ress A nalysis Maximum stress at point P can be summed in the following way: (5.19) (5.20) where I is the second moment of area about the neutral axis, c is the location of the centroid of the beam and A is the area o f the cross section of the beam he n ce:

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36

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37 Chapter 6 Results and Discussion 6.1 Kinematic improvements The spherical PRBM allows that for a given value of aspect ratio, and beam angle, one can find a value of characteristic radius factor, and parametric angle coefficient, that best approximates the motion (position and orientation of beam at various input displacements) u p to the parameterization limit, Moreover improvements to previous work done by Jagirdar in the calculations of parameters , and were made. Figures 6.1, 6.2, a nd 6.3 illustrate the improved calculations for , and respectively, as well as for each aspect ratio. Figure 6.1 The characteristic radius factor, versus the arc angle of the compliant beam,

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38 Figure 6.2 The parametric angle coefficient, versus the arc angle of the compliant beam, Figure 6.3 The parameterization limit as a function of the arc angle subtended by the circular compliant beam,

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39 6.2 Elastic portion In addition, for a given value of aspect ratio, and beam angle, and characteristic radius factor, one can find a value for the stiffness coefficient, that best appro ximates the motion (position, orientation and stiffness o f a beam at various input displacements) up to the parameterization limit, Figure 6.4 shows plotted against the beam angle Furthermore, the individual components of and were calculat ed and are plotted in Figure 6.5 Figure 6.4 v ersu s Lambda

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40 Figure 6. 5 v ersu s Lambda 6.3 Stress analysis On the other hand, for a given value of aspect ratio, and beam angle, and characteristic radius factor, one can find a value for the maximu m stress at a point P on the beam. Maximum stress at point P was summed in the following way: (6.1) where I is the moment of inertia about the neutral axis, c is the location of the centroid of the beam and A is the area of the cross section of the beam, hence:

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41 6.4 Conclusion The kinematics of a compliant curved beam and its rigid body equivalent has been improved. The procedure for analyzing curved compliant b eams in a FEA program was imp roved by writing a batch code Pseudo rigid body parameters were calculated from FEA results. These parameters are the characteristic radius factor, the parametric angle co efficient the parameterization limit s and Additionally, a stress analysis was performed on the beam.

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42 References [1] Salamon, B.A., Design Automation Conference, DE Vol44 2, pp. 47 51. [2] ble Link DET 83. [3] dissertation, Purdue University, West Lafayette, IN. [4] Vol. 23, p. 20, 1960. [5] with on UT. [6] Howel l L. L. and Midha, A., 1996 Closure theory for the analysis and 118, No. 1, pp 121 125. [7] Howell, L. L., 2001 Compliant Mechanisms 1 st Ed. Wiley Interscience, New York, NY. [8] Byrd, P.F, and Fri edman, M. D., 1954 Handbook of Elliptic Integrals for Engineers and Physicists, Springer Verlag, Berlin. [9] End of Mechanical Design, ASME, Vol. 117, No.1, pp. 156 165. [10] Rigid Body Model Concept and its University, Provo, UT. [11] umerical Results from Large Deflection Beam and Frame Problems Analyzed by Means of Elliptic Integrals, Journal for Numerical Methods in Engineering, Vol. 17, pp 145 153.

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43 [12] s of Freedom of Compliant Mechanisms using the Pseudo Proceedings of the 9th World Congress on the Theory of Machines and Mechanisms, Milano, Italy, Vol. 22, pp. 1537 1541. [13] Mohammad H. Dado Variable parametric Pseu do Rigid Body Model for Large deflection beams with end loads linear Mechanics, Vol. 36, pp 1123 1133. [14] Chao Chieh Lan and Kok Meng Lee ., Generalized Shooting Method he 2005 IEEE International Conference on Robotics and Automation Barcelona, Spain, April 2005 [15] and Machines. Vol. 32, No. 1, pp. 17 38. [16] Chiang, C. H., 1992, Spherical kinematics in contrast to planar kinematics, National Taiwan University, Taipei, Taiwan, Mech anical Mach ine Theory v 27 n 3 May 1992 p 243 250. [17] University of South Florida, Tampa, Fl. [18] mechanism: Pseudo Rigid Body Model 07 [19] Spiegel, M. R. and Liu, J., 1999 Mathematical Handbook of Formulas and Tables. McGraw Hill Washington, D.C. [20] Howel l, L. L. and Midha, A., 1994 The development of force deflection Machine Elements and Machine Dynamics : Proceedings of the 1994 ASME Mechanisms Conference, DE Vol. 71. pp 501 508 [21] Paul, B., 1979, Kinematics and Dynamics of Planar Machinery, Prentice Hall, Upper Saddle River, NJ. [22] Boettama.O. and Roth, B., 1979, Theoretical Kinematics Dover New York. [23] using the Pseudo rigid Mechanism and Machine Theory, Vol. 35, No. 1, pp. 99 115.

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44 [24] r Determining Large Deflections in Compliant Mechanisms Subjected to End Journal of Mechanical Design, Trans. ASME, Vol. 120, No. 3, pp. 392 400. [25] Concepts Mechanisms and Robotics Conference, Salt Lake City, UT, Sep. 28 Oct. 2, 2004. [26] Overrunning Clutch with Centrifugal Throw 9, Nov 21. [27] Crane, N.B., Howell L.L., and Weight, B. Design and Testing of a Compliant Floating Opposing Arm (FOA) Centrifugal in Proceedings of 8th International Power Transmission and Gearing Conference, 2000 ASME Design Engineer ing Technical Conferences DETC2000/PTG 14451 [28] Cannon, J.R., Lusk C.P., and Howell, L.L., Contact Element Proceedings of the ASME Mechanisms and Robotics Conference [29] Jensen, B.D., Howell L.L., and Salmon, L.G., Link, In plane, Bistable Compliant Micro Journal of Mechanical Design Trans. ASME, Vol. 121, No. 3, pp. 93 96. [30] Lai, M.W., Rubin, D., Krempl, E., 1993. Introduction to Continuum Mechanics 3rd ed. Butterworth He inemann, Woburn, MA. [31] Ugural C. A, Fenster K. S, 2003 Advance Strength and Applied Elasticity 4 th Ed. Prentice Hall, New Jersey, NJ. [32] Wolfram, Mathworld, 2006, h ttp://mathworld.wolfram.com/SphericalTriangle.htm.

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45 Appendices

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46 Appendix A: DETC paper

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47 Appendix A (C ontinued)

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48 Appendix A (C ontinued)

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49 Appendix A (C ontinued)

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50 Appendix A (C ontinued)

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51 Appendix A (C ontinued)

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52 Appendix A (C ontinued)

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53 Ap pendix A (C ontinued)

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54 Appendix A (C ontinued)

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55 Appendix A (C ontinued)

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56 Appendix A (C ontinued)

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57 Appendix B: ANSYS batch code !************************************ /CONFIG,NRES,10000 !/CWD,'C: \ Documents and Settings \ aleon2 \ Desktop \ Work' !*********** ************************* !**************************************** !******* Set Up Model Variables ********* !**************************************** !*DO,asp, .4,.7,.3 asp =.1 aspect = 10*asp *DO,arclength,1,126,1 this is really the angle subtended by arc !arclength=40 /title,3D Beam Non linear Deflection /PREP7 LCLEAR, ALL LDELE, ALL KDELE, ALL R=100 PI=acos( 1.) T_arclength = R*arclength*PI/180 this is the true arc length of the beam h1=.1*T_arclength make the height and width small in comparison to the length of the beam b1=h1 b2=.1*T_arclength h2=asp*b2 !*********** Area properties ************** A1 = h1*b1 Iy1= 1/12*b1*h1*h1*h1 this is the I value for th e bending direction use it to normalize the forces Iz1= 1/12*h1*b1*b1*b1 E1= 3000000 !************************************

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58 Appendix B (C ontinued) A2= h2*b2 Iz2= 1/12*h2*b2*b2*b2 Iy2= 1/12*b2*h2*h2*h2 E2= 300 !********** Declare an element type: Beam 4 (3D Elastic) ********* ET,1,BEAM4 KEYOPT,1,2,1 KEYOPT,1,6,1 !********** Set Real Constants 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 !************************************ xcoor=R*cos(arclength*PI/180) zcoor=R*sin(arclength*PI/180 ) !********** Create Keypoints 1 throug 7: K(Point #, X Coord, Y Coord, Z Coord) ********* K,1,0,0,0, K,2,xcoor,0,zcoor, K,3,R,0,0, K,4,1.050*R,0,0, K,5,R,.050*R,0, K,6,R,0, .050*R, K,7,.950*R,0,0 !********* Create Beam using Lines a nd an Arc and divide into segments ********* LSTR, 1, 2 LSTR, 3, 4 Draws lines connecting keypoints 1 through 7 LSTR, 3, 5 LSTR, 3, 6 LSTR, 3, 7

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59 Appendix B (C ontinued ) LESIZE,ALL,,,2 Specifies the divisions and spacing ratio on unmeshed lines, 2 divions on lines 1 through 7 except 6 LARC,2,3,1,R, Defines a circular arc LESIZE,6,,,32 Spe cifies the divisions and spacing ratio on unmeshed lines, 32 divions on line 6 !*********** MESH *********** real,1 Use real constant set 1 type,1 Use element type 1 mat,1 use material property set 1 LMESH,1,5 mesh lines 1 5 real,2 Use real constant set 2 type,1 Use element type 1 mat,2 use material property set 2 LMESH,6 mesh line 6 !******* Get Node Numbers at chosen keypoints ******* ksel,s,kp,,1 nslk,s *get,nkp1,node,0,num,max Retrieves a value and s tores it as a scalar parameter or part of an array parameter*********** 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,,5 nslk,s

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60 Appendix B (C ontinued) *get,nkp5,node,0,num,max nsel,all ksel,all FINISH !*********************************************************** !********************** SOLUTION *************************** !*********************************************************** /SOL ANTYPE,0 Specifies the analysis type and restart status and "0" means that it Performs a static analysis. Valid for all d egrees 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 values for nonlinear analyses !**************** ******************** DK,1, ,0, , ,UX,UY,UZ,ROTX,ROTZ, DK,2, ,0, , ,UY, , , !************************************ loadsteps=800 *DO,step,1,loadsteps,1 theta=step*(.8*arclength)/loadsteps !************************************ DK,3,UZ,0 dispx= (R (R*cos(theta*PI/180))) dispy=R*sin(theta*PI/180) DK,3,UX,dispx DK,3,UY,dispy DK,3,ROTZ,theta*PI/180 LSWRITE,step *ENDDO LSSOLVE,1,loadsteps

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61 Appendix B (C ontinued) /S TATUS,SOLU FINISH !***************************** !******* GET RESULTS ********* !***************************** /POST1 *DIM,rotY1,TABLE,loadSteps *DIM,fx1,TABLE,loadSteps Constrains: DK,1, ,0, , ,UX,UY,UZ,ROTX,ROTZ *DIM,fy1,TABLE,loadSt eps *DIM,fz1,TABLE,loadSteps *DIM,mx1,TABLE,loadSteps *DIM,mz1,TABLE,loadSteps *DIM,fy2,TABLE,loadSteps Constrains: DK,2, ,0, , ,UY, , , *DIM,disX3,TABLE,loadSteps *DIM,disY3,TABLE,loadSteps *DIM,disZ3,TABLE,loadSteps *DIM,fz3,TABLE ,loadSteps Constrains: DK,3,UZ,0 *DIM,fx3,TABLE,loadSteps Constrains: DK,3,UX,dispx *DIM,fy3,TABLE,loadSteps Constrains: DK,3,UY,dispy *DIM, mz3,TABLE,loadSteps Constrains: DK,3,ROTZ,theta*PI/180 *D IM,disX5,TABLE,loadSteps *DIM,disY5,TABLE,loadSteps *DIM,disZ5,TABLE,loadSteps *Do,nn,1,loadSteps set,nn *GET,roty,Node,nkp1,ROT,Y *SET,rotY1(nn),roty *GET,forcex,Node,nkp1,RF,FX *SET,fx1(nn),forcex *GET,forcey,Node,nkp1,RF,FY *SET,fy1(nn),forcey *GET,for cez,Node,nkp1,RF,FZ *SET,fz1(nn),forcez *GET,momx,Node,nkp1,RF,MX *SET,mx1(nn),momx *GET,momz,Node,nkp1,RF,MY *SET,mz1(nn),momz

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62 Appendix B (C ontinued) *GET,forcey,Node,nkp2,RF,Fy *SET,fy2(nn),forcey *GET,disX,Node,nkp3,U,x *SET,disX3(nn),disX *GET,disY,N ode,nkp3,U,Y *SET,disY3(nn),disY *GET,disz,Node,nkp3,U,Z *SET,disZ3(nn),disz *GET,forcez,Node,nkp3,RF,FZ *SET,fz3(nn),forcez *GET,forcey,Node,nkp3,RF,FY *SET,fy3(nn),forcey *GET,forcex,Node,nkp3,RF,FX *SET,fx3(nn),forcex *GET,momz,Node,nkp3,RF,MZ *SET,mz3( nn),momz *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 *ENDDO /output,output_arc%arclength%_asp%aspect%,txt,,Append !************************************************* *************** !***************FILE HEADER: BEAM DATA*************************** !**************************************************************** *MSG,INFO,'t','w','R','E','arclength','Iy2' Writes an output message via the ANSYS message subro utine. Max of 8 items per line % 14C % 10C % 10C % 10C % 10C % 8C *VWRITE,h2,b2,R,E2,arclength,Iy2 Writes data to a file in a formatted sequence. Max of 19 items per line %16.8G % 16.8G % 16.8G % 16.8G % 16.8G % 16.8G !************* ************************************************ !**************DISPLACEMENT DATA SET************************** !*************************************************************

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63 Appendix B (C ontinued) *MSG,INFO,'roty1','disX3','disY3','disZ3','disX5','disY5' ,'disZ5' % 10C % 10C % 8C % 8C % 8C % 8C % 8C *VWRITE,rotY1(1),disX3(1),disY3(1),disZ3(1),disX5(1),disY5(1),disZ5(1) %16.8G % 16.8G % 16.8G % 16.8G % 16.8G % 16.8G % 16.8G !************************************************************* !**************RE ACTIONS AT NODE 1**************************** !************************************************************* *MSG,INFO,'fx1','fy1','fz1','mx1','mz1' % 17C % 16C % 15C % 14C % 3C *VWRITE,fx1(1),fy1(1),fz1(1),mx1(1),mz1(1) %16.8G % 16.8G % 16.8G % 16.8G % 16.8G !************************************************************* !**************REACTIONS AT NODE 2**************************** !************************************************************* *MSG,INFO,'fy2' % 8C *VWRITE,fy2(1) %16.8G !*********** ************************************************** !**************REACTIONS AT NODE 3**************************** !************************************************************* *MSG,INFO,'fz3','fx3','fy3','mz3' % 17C % 16C % 16C % 8C *VWRITE,fz3(1),fx3( 1),fy3(1),mz3(1) %16.8G %16.8G %16.8G %16.8G /output FINISH *ENDDO *ENDDO

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64 Appendix C: MATLAB code %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Ansys data analysis file % % For an Ansys batch file % % which produces an output file % % Version 1: May 18,2007 % % Version 2: June 7,2007 % % Version 3: June 12, 2007 % % Version 4: July 15, 2007 % % Version 5: July 20, 2007 % % Version 6: August 9, 2007 % % Version 8: October 20, 2007 % % Ansys Data File must provide: DATA % % rotation @ O: Y in column 1 % % displacements @ Q: X,Y,Z in columns 2, 3 & 4 respectively % % displacement from A2 keypoint to D2 ke ypoint (more DA): % % X,Y,Z in columns 5,6,7 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% clear all R = 100; % radius of sphere = 100 arclength_v = 1:1:112; % define the vector of arclengths %data_range = [2 2 2]; data_range = [112 110 98]; % Each aspect ratio has a different ending data point so this range is needed a_length = max(data_range); asp=[0.1 0.4 0.7]; asp_length = length(asp); CTHET A = zeros(a_length,asp_length); CTHETA_check = zeros(a_length,asp_length); GAMMA = zeros(a_length,asp_length); CAPTHETA = zeros(a_length,asp_length); KM = zeros(a_length,asp_length); KF = zeros(a_length,asp_length); P1 = zeros(a_length,asp_length); P2 = ze ros(a_length,asp_length); P3 = zeros(a_length,asp_length); P4 = zeros(a_length,asp_length); CAPTHETA_MAX_M = zeros(a_length,asp_length); CAPTHETA_MAX_F = zeros(a_length,asp_length); CAPTHETA_MAX_FM = zeros(a_length,asp_length); CAPTHETA_MAX = zeros(a_lengt h,asp_length); X = zeros(a_length,asp_length); % Number of data points W.R.T Beta Y = zeros(a_length,asp_length); % Number of data points W.R.T Gamma % assume a2 is a vector [ 0 5 0] a2 = [0 5 0]; % a2 is length of y frame vector at undeflected posi tion for i=1:asp_length, for counter=1:data_range(i), counter arclength = arclength_v(counter);

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65 Appendix C (C ontinued) lambda = arclength*pi/180; aspect=10*asp(i); filename = [ 'output_arc' ,num2str(arcleng th), '_asp' ,num2str(aspect), '.txt' ]; %string1 = 'C: \ DOCUME~1 \ clusk2 \ 800_Loadsteps \ '; %Dr. Lusk's directory string1 = 'C: \ Docume~1 \ aleon2 \ Desktop \ Work \ Solution \ 1600_Loadsteps_ALL_8_9_07 \ ; % Working at school %string1 = 'C: \ Docume~1 \ Owner \ Desktop \ Solution \ 1600_Loadsteps_ALL \ '; %Working at home %string1 = 'C: \ Docume~1 \ clusk2 \ MyDocu~1 \ Research \ studen~1 \ AlexLe~1 \ 800_Loadsteps \ ; fid1 = fopen([string1,filenam e]); % opens the file ABT = fread(fid1); % reads the file into variable ABT in machine code fclose(fid1); % closes the data file GBT = native2unicode(ABT)'; % changes data from machine code to text and writes it to GBT sB = findstr( 'Iy2' GBT); % finds end of first header sF = findstr( 'roty1' ,GBT); % finds beginning of second header s_iB = findstr( 'disZ5' GBT); % finds end of first header s_iF = findstr( 'fx1' GBT); % finds beginning of second header s_iB2 = findstr( 'mz1' GBT); % finds end of second header s_iF2 = findstr( 'fy2' GBT); % finds beginning of third header %s_iB3 = findstr('mz1', GBT); % finds end of second header s_iF3 = findstr( 'fz3' GBT); % finds beginning of second hea der s_iB3 = findstr( 'mz3' GBT); header = str2num(GBT(sB(end)+4:sF(end) 1)); DATA = str2num(GBT(s_iB(end)+6:s_iF(end) 1)); % turns the data into a numerical matrix % DATA2 = str2num(GBT(s_iB2(end)+4:s_iF2(end) 1 )); % gets the second chunk of data % DATA3 = str2num(GBT(s_iF2(end)+4:s_iF3(end) 1)); DATA4 = str2num(GBT(s_iB3(end)+4:end)); D_size = size(DATA); E = header(4); Iy2 = header(6); t = header(1); w = header(2); Iy1 = 1/12*w^3*t;

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66 Appendix C (C ontinued) phi_c = DATA(:,1); % phi is the rotation of the rigid link in the X Z plane about y beta_c = atan2(D ATA(:,3),R + DATA(:,2)); % beta is defined by the displacement of point Q % fx1 = DATA2(:,1); % fy1 = DATA2(:,2); % fz1 = DATA2(:,3); % mx1 = DATA2(:,4); % mz1 = DATA2(:,5); % fy2 = DATA3; % fz3 = DATA4(:,1); fx3_p = DATA4(:,2); fy3_p = DATA4(:,3); mz3_p = DATA4(:,4); % fx3 = DATA4(:,2); % fy3 = DATA4(:,3); % mz3 = DATA4(:,4); [pfx,sx] = polyfit(beta_c,fx3_p,15); [pfy,sy] = polyfit(beta_c,fy3_p,15); [pfz,sz] = polyfit(beta_c,mz3_p,15); [fx3,dx3] = polyval(pfx,beta_c,sx); [fy3,dy3] = polyval(pfy,beta_c,sy); [mz3,dz3] = polyval( pfz,beta_c,sz); fc1 = fx3.*cos(beta_c)+fy3.*sin(beta_c); fc2 = fx3.*sin(beta_c)+fy3.*cos(beta_c); FC1(counter,i,:) = fc1*(lambda*R)^2/E/Iy2; FC2(counter,i,:) = fc2*(lambda*R)^2/E/Iy2; d2 = ones(D_size(1),1)*a2 + [DATA(:,5),DATA(:,6),DATA(:,7)] [DATA(:,2),DATA(:,3),DATA(:,4)]; mag_d2 = (d2(:,1).^2+d2(:,2).^2+d2(:,3).^2).^.5; da22 = d2(:,2)./mag_d2; da23 = d2(:,3)./mag_d2; theta0_c = atan2(da23 ,da22); % PHI(counter,i,:) = phi_c; % BETA(counter,i,:) = beta_c; % THETA0(counter,i,:) = theta0_c; dgamma = .00005; gamma_r = 0.75:dgamma:.9; gamma = ones(length(phi_c),1)*gamma_r; beta = b eta_c *ones(1,length(gamma_r)); phi = phi_c *ones(1,length(gamma_r)); theta0 = theta0_c *ones(1,length(gamma_r)); gamma_l = gamma*lambda; %phi =PHI(1,countBETA); sincaptheta = sin(beta)./sin(gamma_l); coscaptheta = tan((gamma_l phi)).*cot(gamma_l); captheta = atan2(sincaptheta,coscaptheta); % epsilon_e is FEA based calculation of displacement vector from the original % to the final location of the end of the beam. It is indepe ndent % of the guess values for gamma.

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67 Appendix C (C ontinued) abs_epsilon_e = sqrt((cos(beta).*cos(phi) ones(size(beta))).^2+(sin(beta)).^2+(cos(beta).*sin(phi)).^2); % magnitude of vector epsilon_e epsilon_ex = cos(beta).*cos(p hi) 1; % x component of epsilon_e epsilon_ey = sin(beta); % y component of epsilon_e epsilon_ez = cos(beta).*sin(phi); % z component of epsilon_e % epsilon_a is PRBM based calculation of displacement vector from the original % to the final location of the end of the beam. epsilon_ax = (cos(gamma_l)).^2.*(1 cos(captheta))+cos(captheta) 1; % x component of epsilon_a epsilon_ay = sin(captheta).*sin(gamma_l); % y component of epsilon_a epsilon_az = sin(gamma_l).*cos(gamma_l).*(1 cos(captheta)); % z component of epsilon_a error = sqrt((epsilon_ex epsilon_ax).^2 +(epsilon_ey epsilon_ay).^2 +(epsilon_ez epsilon_az).^2); % magnitude of vector difference o f epsilon_e and epsilon_a rel_error = error./abs_epsilon_e; % bool_rel_error is 1 if rel_error is less than .005 bool_rel_error = 1*floor(.5*(sign(rel_error .005))); d_bool = [zeros(1,length(gamma_r));diff (bool_rel_error)]; flag1 = 1 (cumsum(ceil(.5*d_bool))); bool_rel_error_fixed = flag1.*bool_rel_error; gamma_range_metric = sum(bool_rel_error_fixed); [y,x]=max(gamma_range_metric); Y(counter,i) = y; X(coun ter,i) = x; [c,h] = contour(rel_error,[0.005 0.005]); drawnow a = get(h, 'ContourMatrix' ); ax = a(1,:) x; ay = a(2,:) y; ai = find(abs(ax)<10); [jumpsize, jumpspot] = max(diff(ai)); top_ax = ax(ai(1):ai(jumpspot)); top_ay = ay(ai(1):ai(jumpspot)); side_ax = ax(ai(jumpspot+1):ai(end)); side_ay = ay(ai(jumpspot+1):ai(end)); slope_side = diff(side_ay)./diff(side_ax); [max_slope, max_ slope_spot] = max(slope_side); xa = side_ax(max_slope_spot); % adjustment to x gamma_refine = xa*dgamma; [ati_plus] = find(top_ax>xa); ax1 = top_ax(ati_plus(1)); ay1 = top_ay(ati_plus(1)); [ati_minus] = find(top_ax
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68 Appendix C (C ontinued) ax0 = top_ax(ati_minus(end)); ay0 = top_ay(ati_minus(end)); ya = (ay1 ay0)/(ax1 ax0)*(xa ax0) + ay0 ; % adjustment to y % y is index of the largest value of beta that gives a rel error < % .5%, x is the index of the best value of gamma true_gamma = gamma_r(x) + gamma_refine; % best value of gamma t_gamma_l = true_gamma*lambda; % recalculate captheta with true gamma d_beta = beta_c(y+1) beta_c(y); beta_refine = d_beta*ya; d_phi = phi_c(y+1) phi_c(y); phi_refine = d_phi*ya; true_sincaptheta = sin(beta_c)/sin(t_gamma_l); true_coscaptheta = tan((t_gamma_l phi_c)).*cot(t_gamma_l); captheta_v = atan2(true_sincaptheta,true_coscaptheta); sincaptheta_max = sin(beta_c(y)+beta_refine)/sin(t_gamma_l); coscaptheta_max = tan((t_gamma_l (ph i_c(y)+phi_refine))).*cot(t_gamma_l); captheta_max = atan2(sincaptheta_max,coscaptheta_max); % calculate forces/moments within short range kf_t = (cos(captheta_v).*sin(t_gamma_l))./cos(beta_c).*(( fx3*R.*sin(beta_c) + fy3 *R.*cos(beta_c) ))./captheta_v; km_t = (cos(captheta_v).*sin(t_gamma_l))./cos(beta_c).*(( mz3))./captheta_v; km = polyfit(captheta_v(1:end),km_t(1:end),0)*((lambda*R)/E/Iy2); kf = polyfit(captheta_v(1:end),kf_t(1:end),0)*(( lambda*R)/E/Iy2); km_theta = km/true_gamma; kf_theta = kf/true_gamma; KM(counter,i) = km; KF(counter,i) = kf; KMTHETA(counter,i,:) = km_theta; KFTHETA(counter,i,:) = kf_theta; FX3(counter,i,:) = fx3; FY3(counter,i,:) = fy3; MZ3(counter,i,:) = mz3; %*********** try to figure out fc1 f1 = fc1*(lambda*R)^3/E/Iy2;

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69 Appendix C (C ontinued) [p,s] = polyfit(captheta_v(1:y),f1(1:y),3); % 3rd order fit seems to work best P1(counter,i) = p(1); P2(counter,i) = p(2); P3(counter,i) = p(3); P4(counter,i) = p(4); [yy,delta] = polyval(p,captheta_v,s); % figure(1) % plot( captheta_v*180/pi,[f1,y,y delta,y+delta]) % % %Ro = (f.^2/KR 2*R*f)./(2*KR*(1 cos(beta_c)) 2*f); % % rho = (R^2 2*R*Ro.*cos(beta_c)+Ro.^2).^.5; % %************stress % alpha_c = lambda phi_c; % sigma_x_f1 = fc1.*( cos(beta_c).*sin(alpha_c))*(1/(t*w) R*w/(2*Iy1)); % sigma_x_f2 = fc2.*sin(alpha_c).*(sin(beta_c)./(t*w)+R*t/(2*Iy2)+R*w*sin(beta_c)/(2*I y1)); % sigma_x_mz = mz3.*sin(alpha_c)*t/(2*Iy2); % tau_xy_f2 = R*f c2.*(cos(alpha_c) cos(beta_c))/(.312*w*t^2); % tau_xy_f1 = R*fc1.*sin(beta_c)/(.312*w*t^2); % tau_xy_mz = mz3.*cos(alpha_c)/(.312*w*t^2); % figure(3) % plot(captheta_v*180/pi,[sigma_x_f1+sigma_x_f2+sigma_x_mz,sigma_x_f 1,sig ma_x_f2,sigma_x_mz]) % figure(2) % plot(captheta_v*180/pi,[tau_xy_f1+tau_xy_f2+tau_xy_mz,tau_xy_f1,tau_xy_ f2,tau_xy_mz]) % tmax_f1 = (sigma_x_f1.^2+tau_xy_f1.^2).^.5; % tmax_f2 = (sigma_x_f2.^2+tau_xy_f2.^2).^.5; % tmax_mz = (sigma_x_mz.^2+tau_xy_mz.^2).^.5; % sigma_x = sigma_x_f1+sigma_x_f2+sigma_x_mz; % tau_xy = tau_xy_f1+tau_xy_f2+tau_xy_mz; % tmax = (sigma_x.^2+tau_xy.^2).^.5; % figure(3) % plot(captheta_v*180/pi,tmax. /(tmax_f2+tmax_mz)) % tmax is the maxium shear stress including f1,f2& mz, f1 reduces the stress in some cases. % pause %figure(1) %hold on %plot(captheta_v*180/pi,[km.*captheta_v,(cos(captheta_v).*sin(t_gamma_l ))./cos(beta _c).*(mz3)*((lambda*R)/E/Iy2)]) % error_m =(km.*captheta_v ((cos(captheta_v).*sin(t_gamma_l))./cos(beta_c).*(mz3)*((lambda*R)/E/Iy 2)))./((cos(captheta_v).*sin(t_gamma_l))./cos(beta_c).*(mz3)*((lambda*R )/E/Iy2));

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70 Appendix C (C ontinued) % M_ error_m = [abs(error_m), abs(error_m)] % [c,h] = contour(M_error_m, [0.04 0.04]) % a = get(h,'ContourMatrix'); % ay = a(2,:) % spot_m = max(ay); % ym = floor(spot_m); % incr = spot_m ym; % d_capm = ca ptheta_v(ym+1) captheta_v(ym); % captheta_m_refine = incr*d_capm; % captheta_max_m = captheta_v(ym) +captheta_m_refine; % CAPTHETA_MAX_M(counter,i) = captheta_max_m; % % % figure(2) % %hold on % %plot( captheta_v*180/pi,[kf.*captheta_v,(cos(captheta_v).*sin(t_gamma_l ))./cos(beta_c).*(fc2*R)*((lambda*R)/E/Iy2)]) % error_f = (kf.*captheta_v ((cos(captheta_v).*sin(t_gamma_l))./cos(beta_c).*(fc2*R)*((lambda*R)/E/ Iy2)))./((cos(captheta_v).*sin(t_gamm a_l))./cos(beta_c).*(fc2*R)*((lamb da*R)/E/Iy2)); % M_error_f = [abs(error_f), abs(error_f)] % [c,h] = contour(M_error_f, [0.04 0.04]) % a = get(h,'ContourMatrix'); % ay = a(2,:) % spot_f = max(ay); % yf = floor(spot_f); % incr = spot_f yf; % d_capf = captheta_v(yf+1) captheta_v(yf); % captheta_f_refine = incr*d_capf; % captheta_max_f = captheta_v(yf) +captheta_f_refine; % CAPTHETA_MAX_F(counter,i) = captheta_max_ f; %figure(3) % hold on %plot(captheta_v*180/pi,[kf.*captheta_v,(cos(captheta_v).*sin(t_gamma_l ))./cos(beta_c).*(fc2*R)*((lambda*R)/E/Iy2)]) error_fm = ((kf+km).*captheta_v ((cos(captheta_v).*sin(t_gamma_l))./cos(bet a_c).*(fc2*R+mz3)*((lambda*R )/E/Iy2)))./((cos(captheta_v).*sin(t_gamma_l))./cos(beta_c).*(fc2*R+mz3 )*((lambda*R)/E/Iy2)); M_error_fm = [abs(error_fm), abs(error_fm)] [c,h] = contour(M_error_fm, [0.04 0.04]) a = get(h, 'Contou rMatrix' ); ay = a(2,:) spot_fm = max(ay); yfm = floor(spot_fm); incr = spot_fm yfm; d_capfm = captheta_v(yfm+1) captheta_v(yfm); captheta_fm_refine = incr*d_capfm; captheta_max_fm = captheta_v(yfm) +c aptheta_fm_refine; CAPTHETA_MAX_FM(counter,i) = captheta_max_fm;

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71 Appendix C (C ontinued) %CAPTHETA_MAX(counter,i) = captheta_v(end); % k_resol = (((km+kf).*captheta_v) ((cos(captheta_v).*sin(t_gamma_l))./cos(beta_c).*(fc2* R+mz3)*((lambda*R )/E/Iy2)))./((cos(captheta_v).*sin(t_gamma_l))./cos(beta_c).*(fc2*R+mz3 )*((lambda*R)/E/Iy2)); %min_k = min(k); % min_k = 1.5/((lambda*R)/E/Iy2); % max_k = 2.5/((lambda*R)/E/Iy2); % %max_k = max(k); % dk = (max_k min_k)./1000; % K_r = [min_k:dk:max_k]; % fx_m = fx3*ones(size(K_r)); % fy_m = fy3*ones(size(K_r)); % mz_m = mz3*ones(size(K_r)); % beta_m = beta_c*ones(size(K_r)); % captheta_m = captheta(:,x)*o nes(size(K_r)); % K_m = ones(size(fx3))*K_r; % error2 = fx_m.*( R*tan(beta_m).*cos(captheta_m)*sin(t_gamma_l)) + fy_m.*(R*cos(captheta_m)*sin(t_gamma_l)) + mz_m.*cos(captheta_m)./cos(beta_m).*sin(t_gamma_l) K_m.*captheta_m; % tv = fx_m.*( R*tan(beta_m).*cos(captheta_m).*sin(t_gamma_l)) + fy_m.*(R*cos(captheta_m).*sin(t_gamma_l)) + mz_m.*cos(captheta_m)./cos(beta_m).*sin(t_gamma_l); % rel_error2 = abs(error2./tv); % % bool_rel_error2 = 1*floor(.5*(sign(rel_e rror2 .05))); % d_bool2 = [zeros(1,length(K_r));diff(bool_rel_error2)]; % %flag2 = 1 (cumsum(ceil(.5*d_bool2))); % %bool_rel_error_fixed2 = flag2.*bool_rel_error2; % %K_range_metric2 = sum(bool_rel_error_fixed2); % K_range_metric2 = sum(bool_rel_error2); % [y2,x2]=max(K_range_metric2); % Y2(counter,i) = y2; % X2(counter,i) = x2; % true_k = K_r(x2)*(lambda*R)/E/Iy2 % %k_nd = k*lambda*R/E/Iy2; % nondimensional form of k % %KND(counter,i,:) = [k_nd;1./zeros(length(beta_c) y,1)]; % figure(4) % clf % %[c,h]=contour(captheta*180/pi,gamma,rel_error,[0:.01:.05]) % pcolor(captheta_m*180/pi,K_m,bool_rel_error2),shading flat,colorbar('horiz') % %clabel(c,h) % title([' \ lambda = ',num2str(lambda*180/pi)]) % % figure(5) % % clf % % %[c,h]=contour(captheta*180/pi,gamma,rel_error,[0:.01:.05]) % % pcolor(captheta_m*180/pi,K_m,bool_rel_error_fixed2),shad ing flat % % %clabel(c,h) % % drawnow % figure(6)

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72 Appendix C (C ontinued) % clf % %[c,h]=contour(beta,gamma,rel_error,[0:.01:.05]) % %plot(short_captheta*180/pi,short_theta0_c*180/pi) % % plot(sho rt_captheta*180/pi,k_nd) % plot(beta_c,[dx3,dy3,dz3]); % %clabel(c,h) % drawnow % %pause [p2,s2] = polyfit(captheta_v(1:y),theta0_c(1:y),1); CTHETA(counter,i) = p2(1); CTHETA_check(counter,i) = p2(2); GAMMA(counter,i) = true_gamma; CAPTHETA(counter,i) = captheta_max*180/pi; % end % figure(1) % plot(squeeze(PHI(counter,:,:))'*180/pi) % title(['arclength = ',num2str(arclength)]) % xlabel('l oadsteps') % ylabel(' \ phi') % leg_matrix =[]; % create legend matrix % for i =1:asp_length % leg_matrix = [leg_matrix;'aspect ratio = ',num2str(asp(i))]; % end % legend(leg_matrix,'Location','B est') % figure(2) % plot(squeeze(BETA(counter,:,:))'*180/pi) % title(['arclength = ',num2str(arclength)]) % xlabel('loadsteps') % ylabel(' \ beta') % leg_matrix =[]; % create legend matrix % for i =1:asp_length % leg_matrix = [leg_matrix;'aspect ratio = ',num2str(asp(i))]; % end % legend(leg_matrix,'Location','Best') % figure(3) % plot(squeeze(THETA0(counter,:,:))'*180/pi) % title(['arclength = ',num2str(arclength)]) % xlabel('loadsteps') % ylabel(' \ theta_0') % leg_matrix =[]; % create legend matrix % for i =1:asp_length % leg_matrix = [leg_matrix;'aspect ratio = ',num2str(asp(i))]; % end % legend(leg_m atrix,'Location','Best') % pause end [P1_asp1_fit,s11] = polyfit(arclength_v(1:data_range(1))',P1(1:data_range(1),1),5);

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73 Appendix C (C ontinued) [P1_asp4_fit,s12] = polyfit(arclength_v(1:data_range(2))',P1(1:data_range(2),2),5); [P1_asp7_fit,s13] = polyfit(arclength_v(1:data_range(3))',P1(1:data_range(3),3),5); [P1_1_y,P1_delta_1] = polyval(P1_asp1_fit,arclength_v,s11); [P1_2_y,P1_delta_2] = polyval(P1_asp4_fit,arclength_v,s12); [P1_3_y,P1_delta_3] = polyval(P1_asp7_fit,arclength_v,s13); [P2_asp1 _fit,s21] = polyfit(arclength_v(1:data_range(1))',P2(1:data_range(1),1),5); [P2_asp4_fit,s22] = polyfit(arclength_v(1:data_range(2))',P2(1:data_range(2),2),5); [P2_asp7_fit,s23] = polyfit(arclength_v(1:data_range(3))',P2(1:data_range(3),3),5); [P2_1_y,P2_d elta_1] = polyval(P2_asp1_fit,arclength_v,s21); [P2_2_y,P2_delta_2] = polyval(P2_asp4_fit,arclength_v,s22); [P2_3_y,P2_delta_3] = polyval(P2_asp7_fit,arclength_v,s23); [P3_asp1_fit,s31] = polyfit(arclength_v(1:data_range(1))',P3(1:data_range(1),1),5); [P 3_asp4_fit,s32] = polyfit(arclength_v(1:data_range(2))',P3(1:data_range(2),2),5); [P3_asp7_fit,s33] = polyfit(arclength_v(1:data_range(3))',P3(1:data_range(3),3),5); [P3_1_y,P3_delta_1] = polyval(P3_asp1_fit,arclength_v,s31); [P3_2_y,P3_delta_2] = polyval( P3_asp4_fit,arclength_v,s32); [P3_3_y,P3_delta_3] = polyval(P3_asp7_fit,arclength_v,s33); [P4_asp1_fit,s41] = polyfit(arclength_v(1:data_range(1))',P4(1:data_range(1),1),5); [P4_asp4_fit,s42] = polyfit(arclength_v(1:data_range(2))',P4(1:data_range(2),2), 5); [P4_asp7_fit,s43] = polyfit(arclength_v(1:data_range(3))',P4(1:data_range(3),3),5); [P4_1_y,P4_delta_1] = polyval(P4_asp1_fit,arclength_v,s41); [P4_2_y,P4_delta_2] = polyval(P4_asp4_fit,arclength_v,s42); [P4_3_y,P4_delta_3] = polyval(P4_asp7_fit,arclen gth_v,s43); % for i = 1:asp_length % figure(1) % plot(squeeze(PHI(:,i,:))'*180/pi) % title(['aspect ratio = ',num2str(asp(i))]) % xlabel('loadsteps') % ylabel(' \ phi') % leg_matrix =[]; % create legend matrix % if a_length< 10, % for counter =1:a_length % leg_matrix = [leg_matrix;'arc = ',num2str(arclength_v(counter))]; % end

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74 Appendix C (C ontinued) % legend(leg_matrix,'Location','Best') % end % figure(2) % plot(squeeze(BETA(:,i ,:))'*180/pi) % title(['aspect ratio = ',num2str(asp(i))]) % xlabel('loadsteps') % ylabel(' \ beta') % leg_matrix =[]; % create legend matrix % if a_length<10 % for counter =1:a_length % leg_matrix = [leg_matrix;'arc = ',num2str(arclength_v(counter))]; % end % legend(leg_matrix,'Location','Best') % end % figure(3) % plot(squeeze(THETA0(:,i,:))'*180/pi) % title(['aspect ratio = ',num2str(asp(i))]) % xlabel('loadsteps') % ylabel(' \ t heta_0') % leg_matrix =[]; % create legend matrix % if a_length<10, % for counter =1:a_length % leg_matrix = [leg_matrix;'arc = ',num2str(arclength_v(counter))]; % end % legend(leg_matrix,'Location','Best') % end % % % pause % end % C_M = squeeze(MZ3)./squeeze(FC2); flag1 = sign(GAMMA); figure(1) plot(arclength_v,GAMMA./flag1) xlabel( \ lambda, (deg)' ) ylabel( \ gamma' ) legend( 'Asp_{0.1}' 'Asp_{0.4}' 'Asp_{0.7}' 'Location' 'NorthEastOutside ) p rint gamma dtiff r600 print gamma dps r600 figure(2) plot(arclength_v,CTHETA./flag1) xlabel( \ lambda, (deg)' ) ylabel( 'c_{ \ theta}' ) legend( 'Asp_{0.1}' 'Asp_{0.4}' 'Asp_{0.7}' 'Location' 'NorthEastOutside ) print ctheta dtiff r600 print ctheta dps r600

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75 Appendix C (C ontinued) % figure(2) % plot(arclength_v,CTHETA_check) figure(3) plot(arclength_v,CAPTHETA./flag1) xlabel( \ lambda, (deg)' ) ylabel( \ Theta ( \ gamma), (deg)' ) legend( 'Asp_{0.1}' 'Asp_{0.4}' 'Asp_{0.7}' 'Location' 'NorthEastOutside ) pr int captheta_g dtiff r600 print captheta_g dps r600 figure(4) plot(arclength_v,[KM'./flag1';KF'./flag1']) xlabel( \ lambda, (deg)' ) ylabel( 'K_m, K_f' ) legend( 'K_m Asp_{0.1}' 'K_m Asp_{0.4}' 'K_m Asp_{0.7}' 'K_f Asp_{0.1}' 'K_f Asp_{0.4}' 'K_f Asp_{0.7 }' 'Location' 'NorthEastOutside' ) print KmKf dtiff r600 print KmKf dps r600 figure(5) plot(arclength_v,[P1'./flag1';[P1_1_y;P1_2_y;P1_3_y]./flag1']) xlabel( \ lambda, (deg)' ) ylabel( 'P_1' ) legend( 'Data Asp_{0.1}' 'Data Asp_{0.4}' 'Data Asp_{0.7}' 'Curve Fit Asp_{0.1}' 'Curve Fit Asp_{0.4}' 'Curve Fit Asp_{0.7}' 'Location' 'NorthEastOutside' ) print P1 dtiff r600 print P1 dps r600 figure(6) plot(arclength_v,[P2'./flag1';[P2_1_y;P2_2_y;P2_3_y]./flag1']) xlabel( \ lambda, (deg)' ) ylabel( 'P_2' ) l egend( 'Data Asp_{0.1}' 'Data Asp_{0.4}' 'Data Asp_{0.7}' 'Curve Fit Asp_{0.1}' 'Curve Fit Asp_{0.4}' 'Curve Fit Asp_{0.7}' 'Location' 'NorthEastOutside' ) print P2 dtiff r600 print P2 dps r600 figure(7) plot(arclength_v,[P3'./flag1';[P3_1_y;P3_2_y;P 3_3_y]./flag1']) xlabel( \ lambda, (deg)' ) ylabel( 'P_3' ) legend( 'Data Asp_{0.1}' 'Data Asp_{0.4}' 'Data Asp_{0.7}' 'Curve Fit Asp_{0.1}' 'Curve Fit Asp_{0.4}' 'Curve Fit Asp_{0.7}' 'Location' 'NorthEastOutside' )

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76 Appendix C (C ontinued) print P3 dtiff r600 print P3 dps r600 figure(8) plot(arclength_v,[P4'./flag1';[P4_1_y;P4_2_y;P4_3_y]./flag1']) xlabel( \ lambda, (deg)' ) ylabel( 'P_4' ) legend( 'Data Asp_{0.1}' 'Data Asp_{0.4}' 'Data Asp_{0.7}' 'Curve Fit Asp_{0.1}' 'Curve Fit Asp_{0.4}' 'Curve Fit Asp_{0 .7}' 'Location' 'NorthEastOutside' ) print P4 dtiff r600 print P4 dps r600 % figure(5) % plot(arclength_v,P1) % % figure(6) % plot(arclength_v,P2) % % figure(7) % plot(arclength_v,P3) % % figure(8) % plot(arclength_v,P4) % figure(9) % plot(arcl ength_v,CAPTHETA_MAX_M*180/pi./flag1) % % figure(10) % plot(arclength_v,CAPTHETA_MAX_F*180/pi./flag1) figure(11) plot(arclength_v,CAPTHETA_MAX_FM*180/pi./flag1) xlabel( \ lambda, (deg)' ) ylabel( \ Theta (K), (deg)' ) legend( 'Asp_{0.1}' 'Asp_{0.4}' 'Asp_{0. 7}' 'Location' 'NorthEastOutside ) print capthetaK dtiff r600 print capthetaK dps r600 figure (12) plot(captheta_v*180/pi,squeeze(FC1([1,20,45,90,110],1,:))) xlabel( \ Theta, (deg)' ) ylabel( 'F_{c1}' ) title( 'For aspect ratio 0.1' ) text(75, 0.1, \ lambda=1' ) text(65,0.35, \ lambda=20' ) text(35,0.3, \ lambda=45' ) text(81,4, \ lambda=90' ) text(81.16,2.375, \ lambda=110' ) print ForceC1_aps1 dtiff r600

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77 Appendix C (C ontinued) print ForceC1_asp1 dps r600 figure (13) plot(captheta_v*180/pi,squeeze(FC1 ([1,20,45,90,110],2,:))) xlabel( \ Theta, (deg)' ) ylabel( 'F_{c1}' ) title( 'For aspect ratio 0.4' ) text(75, 0.1, \ lambda=1' ) text(79.32,1.184, \ lambda=20' ) text(65.22,0.9737, \ lambda=45' ) text(80.77,4.395, \ lambda=90' ) text(80.15,2.254, \ lambda=110' ) print Fo rceC1_asp4 dtiff r600 print ForceC1_asp4 dps r600 figure (14) plot(captheta_v*180/pi,squeeze(FC1([1,20,45,90,110],3,:))) xlabel( \ Theta, (deg)' ) ylabel( 'F_{c1}' ) title( 'For aspect ratio 0.7' ) text(75.38,0.2588, \ lambda=1' ) text(78.08,1.18, \ lambda=20 ) text(80.56,3.186, \ lambda=45' ) text(80.77,5.007, \ lambda=90' ) text(73.1, 0.1506, \ lambda=110' ) print ForceC1_asp7 dtiff r600 print ForceC1_asp7 dps r600

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78 Spherical Triangles Figure D 1: Spherical triangles A spherical triangle is a figure formed on the surface of a sphere by three great circular arcs inters ecting pair wise in three vertices. The spherical triangle is the spherical analog of the planar triangle and is sometimes called an Euler triangle [32] Let a spherical triangle have angles A, B, and C (measured in radians at the vertices along the surfa ce of the sphere) and let the sphere on which the spherical triangle sits have radius R [32] Napier Rules beam. The derivation of parameters can be easily obtained from two simple rules discovered by John Napier (1550 1617), the inventor of logarithms. (http://www.angelfire.com/nt/navtrig/B2.html). As the right angle does not enter into the

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79 Appendix D (C ontinued) formulas, only five parts are considered. These ar e a, b, and the complements of A, B, and C (or 90 A, 90 B, 90 c) which can be written A', B', and c'. If these five parts are arranged in the order in which they occur in the triangle, any part may be selected and called the middle part; then the two pa rts next to it are called adjacent parts, and the other two are called opposite parts. Figur e D 2 : Five parts arranged in order of occurrence 1. The sine of the middle part equals the product of the tangents of th e adjacent parts. 2. The sine of the middle part equals the product of the cosines of the opposite parts. The right spherical triangle for the PRBM has the sides, The right angle lies is the pseudo rigid body ang le. is the angle opposite to 3

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80 Appendix D (Continued) Figure D 3 : Spherical right triangle

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81 Appendix D (Continued) Figure D 4 : Five parts for PRBM right spherical triangle. Using Napier Rules the following equati ons can be obtained. Where and To get

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82 Appendix D (C ontinued) At =90 o this equation fails to give a va lue of pseudo rigid body angle to overcome th is, is also expressed in an alternate form. From Napier Rules we get and To get