USF Libraries
USF Digital Collections

Design and modeling of a bistable spherical compliant micromechanism

MISSING IMAGE

Material Information

Title:
Design and modeling of a bistable spherical compliant micromechanism
Physical Description:
Book
Language:
English
Creator:
Choueifati, Joseph Georges
Publisher:
University of South Florida
Place of Publication:
Tampa, Fla.
Publication Date:

Subjects

Subjects / Keywords:
Bistable
Compliant
Spherical
Dissertations, Academic -- Mechanical Engineering -- Masters -- USF   ( lcsh )
Genre:
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
ABSTRACT: Compliant bistable mechanisms are mechanisms that have two stable equilibrium positions within their range of motion. Their bistability is mainly due to the elasticity of their members. This thesis introduces a new type of bistable micromechanisms, the Bistable, Spherical, Compliant, four-bar Micromechanism (BSCM). Theory to predict bistable positions and configurations is also developed. Bistabilty was demonstrated through testing done on micro-prototypes. Compared to the mathematical model of the BSCM, Finite element models of the BSCM indicated important qualitative differences in the mechanism's stability behavior and its input-angle-input torque relation. The BSCM has many valuable features, such as: Two stable positions that require power only when moving from one stable position to the other, accurate and repeatable out-of-plane motion with resistance to small perturbations. The BSCM may be useful in several applications such as active Braille systems and Digital Light Processing (DLP) chips.
Thesis:
Thesis (M.S.)--University of South Florida, 2007.
Bibliography:
Includes bibliographical references.
System Details:
System requirements: World Wide Web browser and PDF reader.
System Details:
Mode of access: World Wide Web.
Statement of Responsibility:
by Joseph Georges Choueifati.
General Note:
Title from PDF of title page.
General Note:
Document formatted into pages; contains 112 pages.

Record Information

Source Institution:
University of South Florida Library
Holding Location:
University of South Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 001935151
oclc - 225866603
usfldc doi - E14-SFE0002290
usfldc handle - e14.2290
System ID:
SFS0026608:00001


This item is only available as the following downloads:


Full Text

PAGE 1

Design and Modeling of a Bistable S pherical Compliant Micromechanism by Joseph Georges Choueifati A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Department of Mechanical Engineering College of Engineering University of South Florida Major Professor: Craig Lusk, Ph.D. Daniel Hess, Ph.D. Jose Porteiro, Ph.D. Date of Approval: November 7, 2007 Keywords: (Bistable, spherical, compliant) Copyright 2007, Jos eph Georges Choueifati

PAGE 2

Dedication To my Father, Georges, my Mother, Amal, my Brother, Jules, and my Sister, Joelle.

PAGE 3

Acknowledgments I would like to express my gratitude to my major professor Dr. Craig Lusk for his continuous support during this j ourney. He has been more than a mentor; he has been a good friend. His patience and expertise were key factors in the completion of this thesis. It was a gr eat privilege working under his direction. I would like to thank my graduate commi ttee for their time spent on my behalf: Dr. Hess, and Dr. Porteiro. Thei r guidance was invaluable. Fellow graduate students and friend s Alejandro Leon, Carlos F. Acosta, Vivek Ramadoss, Christopher Cheatham, Michael Nellis and Antoine Awwad for the moral support provided. Special recognition should be given to my family for supporting and motivating me through it all. Without them, I would not be where I am today.

PAGE 4

i Table of Contents List of Tables iii List of Figures iv Abstract vi Chapter 1 1 1.1 Objective 1 1.2 Motivation 2 1.3 Contribution 2 1.4 Research Approach 2 Chapter 2: Introduction 3 2.1 MEMS History 3 2.2 Background 4 2.2.1 Surface Micromachining 4 2.2.2 Ortho-Planar Mechanisms 5 2.2.3 Spherical Mechanisms 6 2.2.4 Compliant Mechanisms 7 2.2.5 Bistable Mechanisms 8 Chapter 3: Mathemat ical Background 9 3.1 Planar Mechanisms 9 3.1.1 Positions Analysis of Planar Four-Bar Mechanism 9 3.1.1.1 Closed-Form Equations 10 3.1.2 Pseudo-Rigid-Body Model 11 3.1.2.1 Small Length Flexural Pivot 12 3.1.2.2 PRBM Four-Bar Mechanism 14 3.1.3 Definition of Bistability 16 3.2 Spherical Mechanisms 17 3.2.1 Spherical Trigonometry 18 3.2.2 Spherical Four-Bar Mechanism 21 Chapter 4: Bistability of a Spherical Four-Bar Mechanism 24 4.1 Principle of Virtual Work 24 4.2 Virtual Work Equation of Compliant Spherical Four-Bar Mechanism 25

PAGE 5

ii 4.3 A Simplified Mathematical Model of A Bistable CSM 29 4.3.1 Joint 1: The Flexural Pivot Connecting Links r1 and r2 30 4.3.2 Joint 2: The Flexural Pi vot Connecting Links r2 and r3 31 4.3.3 Joint 3: The Flexural Pi vot Connecting Links r3 and r4 34 4.3.4 Joint 4: The Flexural Pi vot Connecting Links r4 and r1 36 Chapter 5: A Bistable Spherical Compliant Micromechanism 39 5.1 Fabrication Process 39 5.2 Design 41 5.3 BSCM Testing 44 5.4 Finite-Element Analysis 45 Chapter 6 48 6.1 Conclusions 48 6.2 Recommendations for Future Works 48 References 50 Appendices 53 Appendix A: ANSYS Batch Files 54 Appendix B: Output ANSYS Text Files 83 Appendix C: MatLab M-File 112

PAGE 6

iii List of Tables Table 5.1 Layer Names, Thicknesses and Lithography Levels 40 Table 5.2 Minimum Feature Size per Layer of Polysilicon 41

PAGE 7

iv List of Figures Figure 2.1 Ortho-Planar Spherical Mechanism 5 Figure 2.2 Spherical Bistable Mechanism 5 Figure 2.3 Compliant MEMS 8 Figure 3.1 Rigid Link Four-Bar Crank-Rocker Mechanism 10 Figure 3.2 Small-Length Flexural Pivot 12 Figure 3.3 Pseudo-Rigid-Body Model 13 Figure 3.4 A Compliant Four-Bar Mechanism and its Pseudo-Rigid body model 15 Figure 3.5 Ball-On-The-Hill Analogy for Bistable Mechanism 17 Figure 3.6 Spherical Mechanism 18 Figure 3.7 A Spherical Triangle with Sides a,b,c and Dihedral Angles A,B,C 19 Figure 3.8 Schematic of the Part s of Right Spherical Triangle with Right Angle “C” for Use with Napier's Rules 21 Figure 3.9 Spherical Four Bar Mechanism with Links r1, r2, r3 and r4. 22 Figure 4.1 Compliant Spherical Four-B ar Mechanism and Its PRBM 26 Figure 4.2 PRBM of A Spherical Mec hanism with Known Link Lengths 29 Figure 4.3 Input Moment Required to Hold the Spherical Four-Bar Mechanism in Equilibrium from Joint 1 31 Figure 4.4 Input Moment Required to Hold the Spherical Four-Bar Mechanism in Equilibrium from Joint 2 33

PAGE 8

v Figure 4.5 Input Moment Required to Hold the Spherical Four-Bar Mechanism in Equilibrium from Joint 3 35 Figure 4.6 Input Moment Required to Hold the Spherical Four-Bar Mechanism in Equilibrium from Joint 4 37 Figure 5.1 Cross Sectional View S howing all 7 Layers of the MUMP 40 Figure 5.2 BSCM Drawing 42 Figure 5.3 Scanning Electron Microscope (SEM) Image of BSCM as Fabricated: First Stable Equilibrium Position 43 Figure 5.4 SEM Image of the Staple Hinge 43 Figure 5.5 SEM Image of the BSCMEMS in an Out-Of-Plane Position 44 Figure 5.6 SEM Images of BSCM in its Second St able Equilibrium Position 45 Figure 5.7 Mathematical Model of the Moment-Rotation Relationship for a BSCM with Rigid Links and ShortLength Flexural Pivots 46 Figure 5.8 Moment-Rotation Relationshi p from a FEA of the BSCM Prototype 46

PAGE 9

vi Design and Modeling of a Bistable Spherical Compliant Micromechanism Joseph Georges Choueifati ABSTRACT Compliant bistable mechanisms are mechanisms that have two stable equilibrium positions within their range of mo tion. Their bistability is mainly due to a combination of the elasticity of t heir members and their force transmission properties. This thesis introduces a new type of bistable micromechanisms, the Bistable, Spherical, Compliant, fourbar Micromechanism (BSCM). Theory to predict bistable positions and configurations is also developed. Bistabilty was demonstrated through testing done on mi cro-prototypes. Compared to the mathematical model of t he BSCM, Finite element m odels of the BSCM indicated important qualitative differ ences in the mechanism’s stability behavior and its input-angle-input torque relation. The BSC M has many valuable features, such as: two stable positions that require power only when moving from one stable position to the other, prec ise and repeatable out-of-pl ane motion with resistance to small perturbations. The BSCM may be useful in several applications such as active Braille systems and Digital Light Processing (DLP) chips.

PAGE 10

1 Chapter 1 1.1 Objective The objective of this research wa s to develop a bistable Micro-electromechanical system with precise and repeat able out of plane motion. Combining spherical mechanism theory and compliant mechanism theory, large out of plane motion can be achieved. To insure its pr ecision and repeatability, the mechanism was designed to have two st able equilibrium positions. The theory developed was demonstr ated through testing performed on macro and micro-scaled devices that were designed, fabricated and analyzed as part of this research. Finite-element analysis (FEA) was used to predict the motion and bistable behavior of the mechan ism. Possible applications for such a mechanism were also considered. 1.2 Motivation The most common technique used in building MEMS is surface micromachining [1, 2] because of its simpli city and low cost. A challenge in using surface micromachining is t hat the process produces e ssentially two dimensional products. The ratio of the length and width with respect to the thickness of the

PAGE 11

2 elements created is high, thus most MEMS have a planar working space, where the motion of their links traces a single pla ne, [3]. In some applications such as active Braille [4], micro-optic al systems [5], it may be us eful for MEMS to achieve accurate three-dimensional motion. 1.3 Contribution The research in this thesis provides a new design in the MEMS field. A bistable spherical compliant four-bar micromechanism. The theory for modeling the mechanism is presented. Microprotot ypes were designed and fabricated using the Multi Users MEMS Process (MUM Ps) to insure feasibility and theory verification. For further understanding of the mechanism’s behavior several models were developed. 1.4 Research Approach The next chapter, Chapt er 2, provides a background on MEMS and MEMS applications and t heir impact on today’s technology. In Chapter 3 mathematical background useful in descr ibing the mechanisms is presented. Chapter 4 presents the mathematical mo del of a bistable spherical compliant four-bar mechanism (BSCM). In Chapter 5 a MEMS prototype of the BSCM is presented and test results are discusse d with an FEA analysis of several configurations of the ME MS prototype. Chapter 6 provides conclusions and future recommendations.

PAGE 12

3 Chapter 2 Introduction 2.1 MEMS History For the past two decades, integrated circuit technology has enabled researchers and scientists to create microscaled machines that can interact with their environment mechanically and electronically on the micro scale. These systems are called Microelectromechanica l Systems (MEMS) and are being used in several industrial areas and are being in tegrated in devices that are used in our everyday life. One of the most succe ssful commercialization stories of MEMS is that of the airbag accelerometer. In a four year period, form 1995 to 1999, the airbag accelerometer market share skyrocketed from 20% to 80%, and saved automobile manufacturers 50$ per car [6]. Another prominent use of MEMS is the Texas Instruments Digital Light Pr ocessor (DLP) chip, which the core component of the high definition Samsung DLP projection TV s currently being sold on the market. The DLP chip is probabl y the world's most sophisticated light switch. It contains a rectangular array of up to 2 million hinge-mounted digital micro-mirror devices (DMD); each of t hese micro-mirrors measures less than one-fifth the width of a human hair. Thes e DMD are manipulated by MEMS with accurate but limited out of plane motion [7]. Moreov er, because of their small dimensions, power requirements to activate MEMS are small and usually fall on

PAGE 13

4 the mW scale. Predicted future applicat ions of MEMS range from the aerospace industry, where MEMS will be integrated in navigation systems, to the medical industry where micro-manipulator may oper ate on cells [1]. Briefly, a typical MEMS device can be defined as: (1) device consisting of micro-mechanisms and/or microelectronics, (2) a device t hat can be batch fabricated, and (3) a device that does not require a great deal of assembly to utilize it s functionality. [8] 2.2 Background 2.2.1 Surface Micromachining The most common technique used in building MEMS is surface micromachining because of its simp licity and low cost. In surface micromachining, the silicon wafer acts as the substrate, on whic h multiple layers of thin layers of polysilicon or silicon nitr ide are built [1, 2]. A significant challenge of using surface micromachining, is that it is a two dimens ional process. The elements created can measure in length and width several hundred microns but in thickness less than 10 microns, making them relatively planar. Thus most MEMS have a planar working space [3]. For many of the applications mentioned above it may be necessary for MEMS to achieve three-dimensional motion. To achieve that goal, researchers are requir ed to come up with creative designs. 2.2.2 Ortho-Planar Mechanisms Ortho-planar (OP) mechanisms [9] ar e a type of mechanism that can achieve out-of-plane motion. Ort ho-Planar mechanisms are defined as

PAGE 14

5 mechanisms that are built with their links lo cated in a single plane and motion out of that plane [9]. In his dissertation, Lu sk describes numerous types of OP MEMS and spherical MEMS. See exampl es in Figures 2.1 and 2.2 Figure 2.1: Ortho-Planar Spherical Mechanism [3] Figure 2.2: Spherical Bistable Mechanism [3]

PAGE 15

6 2.2.3 Spherical Mechanisms A spherical mechanism is a mechanism where the axes of rotation of all its revolute joints intersect at a single point. The shortest distance between this point and any of the mechanism’s joints corresponds to the radius of a sphere that is the virtual works pace of that mechanism. Furt her information on spherical mechanisms is given in Chapters Three and Four. One particular application where the us e of spherical MEMS with accurate out-of plane motion might be important is in DLP chips. Indeed, DMDs are tilted back and forth into their ON/OFF positions by MEMS. The maximum tilt angle reached at the present is 12 degrees. According to Texa s Instruments, the higher the tilt angle, the better the resoluti on. Another application where spherical MEMS may be important is in Micro I nput Devices Systems (MIDS). MIDS can be integrated in braille-based typing system that intera cts differently with each finger motion pattern. The computer will then translate different patterns into words [4]. In order to achieve these results, des igns are needed that allow for rapid, large and accurate spatial positioning of arrays of micro-mirrors. According to Fukushige: If a long stroke in the out-of-plane direction, a large output force, and high integration can be simu ltaneously realized, microoptical systems such as actuation of micro-mirrors become possible" [5]. This thesis offers a detailed analysis of a MEMS device with large displacement and precise out-of-plane motion.

PAGE 16

7 2.2.4 Compliant Mechanisms With the manufacturing techniques available at their hands and an understanding of MEMS challenges, resear chers are able to develop elastically deformable micro structures [10]. Mechani sms that rely on elastic deformation of their flexural members to carry out mechanical tasks of transforming and transferring energy forc e and motion are called compliant mechanisms [10]. Furthermore, compliant mechanisms co mbine energy storage and motion, thus eliminating the need for s eparate components of joints and springs [10]. Many products currently on the market such as nail clippers, shampoo caps and mechanical pens make use of compliant s egments in their designs. In addition, studies have shown that one of the main reasons behind the failure of MEMS is joints wear [11]; and that replacing these rigid multi-pieces joints with compliant single member joint will likely increase their lifespan [12,13, 14]. On the other hand; the advantages offered by compliant micromechanisms don’t come without challenges. Their dynamic and kinemati cs analysis is difficult but can be simplified using easier techniques such as the Pseudo-rigid-body model (PRBM) [14]. PRBMs model compliant mechanisms as rigid-body mechanisms. They can predict with high accuracy the non linear large deflections of flexible segments [15]. Chapter 3 offers a mo re elaborate background on PRBM. The use of the PRBM will be discussed in more depth in Chapter 3.

PAGE 17

8 Figure 2.3: Compliant MEMS 2.2.5 Bistable Mechanisms Reduction of power consumptions in MEMS design can lower operating cost and improve performance thus enabling uses in new applic ation. One way of achieving that goal is by using bistabilit y. A bistable mechanism has two stable positions at the two extrem es of its range of motion; it requires low input power because power is only supplied when swit ching the mechanism from one stable state to the other. In this thesis, I will focus on the design and analysis of spherical compliant bistable four-bar MEMS with precise and large out of plane displacement.

PAGE 18

9 Chapter 3 Mathematical Background Several planar compliant bistable mechanisms designs have been demonstrated previously. [16,17,18,19,20]. In this chapter some mathematical concepts implemented in previo us work are described that will be useful in the analysis of a spherical compliant bistable four-bar mechanism. 3.1 Planar Mechanisms This section provides a concise review of the position analysis of a rigidbody mechanism to serve as reference fo r later comprehension of the concepts behind a spherical four-bar mechanism Usually, when analyzing a rigid-body mechanism, it is assumed that the elasti c deformation of the rigid links of that mechanism is negligible relatively to its general motion. In the following section, the position analysis of a four-bar mechanism is presented [21]. 3.1.1 Position Analysis of Planar Four-Bar Mechanism Many methods have been developed fo r the position and displacement analysis of a four-bar mechanism. In this paper, we will focus on the analytical

PAGE 19

10 method, specifically a closed-form so lution for a four-bar crank-rocker mechanism. Figure 3.1: Rigid Link Four-Bar Crank-Rocker Mechanism [15] 3.1.1.1 Closed-Form Equations Consider the four-bar crank-rocker mechanism shown in Figure 1.1. The crank angle,2 is considered to be the input. Using the law of cosines one can find the closed form equations that govern t he position of the mechanism. Using the variables defined in Figure 3.1, the closed form equations are: 2 2 1 2 2 2 1cos 2 (3.1) 1 2 2 2 2 1 12 cos (3.2)

PAGE 20

11 3 2 4 2 2 3 12 cos (3.3) 4 2 3 2 2 4 12 cos (3.4) Considering 20, only two possible solutions exist for each of the angles 3 and4 ; the leading and the lagging solutions. The leading form for each angle is 3 (3.5) 4 (3.6) The lagging form for each angle is 3 (3.7) 4 (3.8) 3.1.2 Pseudo-Rigid-Body Model Numerous methods have been develope d to analyze large deflections [22]. The pseudo-rigid-body model (PRBM) provides a simple approach for the analysis of systems that undergo la rge non-linear elastic deflections [23,24,25,26,27]. This method is very useful when designing compliant mechanisms. Compliant members that undergo large deflections are modeled using rigid-body components with similar fo rce-deflection characteristics [15]. Different types of mechanisms require diffe rent models; in this thesis we the small length flexural pivot (SLFP) and th e four-bar PRBM. Also, the method of virtual work is a fundamental tool for force deflection behavior of a mechanism.

PAGE 21

12Thus, it can be used to determine the force-deflection behavior and hence the bistability of a mechanism [18]. 3.1.2.1 Small-Length Flexural Pivot Consider the cantilever beam shown in Figure 3.2 with a force load F at its end. The beam is deflected by an angle Figure 3.2: Small-Length Flexural Pivot [15] The beam is composed of two segm ents: A small flexible segment “l” called small length flexural pi vot and a large rigid segment “L”. Since “l” is flexible and “L” is rigid, then: L lEI EI (3.8) Where E the material’s Young’s modulus and I the second moment of area.

PAGE 22

13Since the flexible section is a lot s horter than the rigid one, the beam’s motion can be modeled as two rigid links connected by a pin joint. The location of the pin joint would be located at the cent er of the flexural pivot as shown in Figure 3.3. Figure 3.3: Pseudo-Rigid-Body Model The x and y coordinates of the beam’s end are approximated as cos 2 2 l L l a (3.9) And sin 2 l L b (3.10)

PAGE 23

14The beams resistance to deflection is modeled using a torsional spring located at the characteri stic pivot with constant The torque required to deflect the spring of an angle is: K T (3.11) The strain energy stored in the spring is: 2 02 K V (3.12) Where K is the spring constant and is equal to: l EI Kl (3.13) Where lEI designated the stiffness of t he short compliant section. 3.1.2.2 PRBM Four-Bar Mechanism A four-bar mechanism with compliant joints can be modeled using the PRBM concept. Figure 3.4 shows a comp liant four-bar mechanism with its pseudo-rigid-body model. The compliant joints in Figure 3.4-a) are replaced by torsional springs in Figure 3.4-b). The energy equations governing a comp liant four-bar mechanism can be described using the principle of virtual work Virtual work is the result of forces acting on system through a virtual displace ment. A virtual displacement is an assumed infinitesimal change in the positio n coordinates of a system such that the constraints remain satisfied. In the ca se of the PRBM of Fi gure 3.3, the virtual

PAGE 24

15work done by the torsional spring can be derived from the derivative of the potential energy,V with respect to a generalized coordinate, q, [15]. q dq dV W (3.14) a) b) Figure 3.4: Part a) a Compliant Four-Bar Mechanism and Part b) its PseudoRigid Body Model

PAGE 25

16The total energy stored in the mechanism when an input torque inT is applied to link 2r is equal to the sum of the potential energy stored in each torsional spring. 2 4 4 2 3 4 3 2 3 2 2 2 2 12 2 2 2 K K K K VT (3.15) The virtual work can be found also by taking the derivative of TV with respect to2 2 dq dV WT (3.16) 3.1.3 Definition of Bistability As mentioned Chapter 2, a bistable mechanism is a mechanism that has two stable equilibrium point s within its range of mo tion. A mechanism is considered to be in stable equilibrium if it returns to its equilibrium position after being subject to small forces or distur bances. A mechanism is in an unstable equilibrium when a small force causes the mechanism to change positions, usually to a position of stable equilibri um. According to Lagrange-Dirichlet theorem, an object is in a stable equilibr ium when its potential energy is at its local minimum. The bistability concept can be demonstrated with the ball-on-thehill analogy Figure 3.5. A sm all impulse applied on to t he ball at eit her ‘A’ or ‘D’ will make oscillate but then it will settle back into its original position. Positions ‘A’ and ‘D’ would then be considered as stable equilibrium positions, locations where

PAGE 26

17the ball has lowest pot ential energy. At position ‘B’, the ball is considered to be in unstable equilibrium position because unde r a small disturbance the ball is going to go to either ‘A’ or ‘D’ positions. Figure 3.5: Ball-On-The-Hill Analogy for Bistable Mechanisms 3.2 Spherical Mechanisms A spherical mechanism is a mechanism where the axes of rotation of all its revolute joints intersect at a singl e point. The shortest distance between this point and any of the mechanism’s joints corresponds to the radius of a sphere that is the virtual workspace of that mechanism. See Figure 3.6. Unstable equilibrium position Stable equilibrium positions Neutrally Stable

PAGE 27

18 Figure 3.6: A Spherical Mechanism This section also provides a brief over view of spherical kinematics to help those with prior knowledge of planar kinem atics understand spherical kinematics. Planar kinematics can be related to spher ical kinematics by considering a plane as a sphere with infinite radius. 3.2.1 Spherical Trigonometry The concise analysis of spherical tr igonometry given here is based on Spiegel and Liu and develops analogies between spherical trigonometry and plane trigonometry [28]. In planar tri gonometry, relationships between straight lines, angles and triangles are obtained on the surface of a flat plane. In spherical trigonometry, the surface is no longer flat but curved according to the surface of the sphere. Thus, geometrical figures are no longer planar but can have mathematically similar properties to t heir planar counterparts. Circles with the same radius of the sphere that are drawn on the surface of the sphere are called great circles. A great circle displays similar mathematical properties as a straight

PAGE 28

19line in a plane. Arcs that bel ong to great circles are call ed great arcs. Each great circle is contained in a pla ne that intersects the sphere. The normal to that plane passing through the center of the sphere is the pole of the great circle. The intersection of two great circles of the same sphere form what is called the dihedral angle. A spherical triangle is a triangle form ed by the intersection of three great circles with its sides bei ng three great arcs and its angles three dihedral angles. Just as for a planar triangl e, there is a Law of Sines and Laws of Cosines that can be applied [29]. In this thesis, upper-case roman le tters represent the dihedral angles between the two planes containing inters ecting great circles; lower-case roman letters represent great arcs. Consider the followi ng spherical triangle ABC on the sphere shown in Figure 3.4. Figure 3.7: A Spherical Triangle with Sides a, b and c and Dihedral Angles A, B and C.

PAGE 29

20 In spherical trigonometry there ar e two Laws of Cosines. The first one relates one dihedral angle and three arcs: A c b c b a cos sin sin cos cos cos (3.17) The second Law of Cosines relate s one arc with three dihedral angles: a C B C B A cos sin sin cos cos cos (3.18) The spherical Law of Sines relates tw o arcs and their opposite two dihedral angles: C c B b A a sin sin sin sin sin sin (3.19) If one of the dihedral angles of t he spherical triangle is equal to 2 then the triangle is a right spherical triangle and Napier’s rules become applicable: The sine of any middle part equals the product of the tangents of the adjacent parts. The sine of any middle part equals the product of the cosines of the opposite parts. There is a simple way to determine which angles are the opposite and which angles are the adjacent angles. Cons ider the spherical triangle of Figure 3.7 in which the dihedral angle “C” is 90 The other 5 angles can be drawn into a circle which has been divided into 5 arcs seen in Figure 3.7.

PAGE 30

21 Figure 3.8: Schematic of the Parts of Right Spherical Triangle with Right Angle “C” for Use with Napier's Rules. In Figure 3.8, A B and c represent the complements of the angles A B and c of the triangle in Figure 3.6. The complement of an angle A is defined by: A = 900-A (3.20) The second rule can be used to find a if A and c are known. The Napier circle shows that the two opposite angles to a are A and c while the two adjacent angles are b and B Then: c A c A a sin sin cos cos sin (3.21) 3.2.2 Spherical Four-Bar Mechanism A spherical four-bar mechanism is a mechanism where the axes of rotation of all pin joints intersect at one point. This point repr esents the center of the sphere that is the virtual workspace of the mechanism.

PAGE 31

22 The spherical trigonometry properties overviewed in the section above can be utilized to develop the mathematical properties of a spherical four bar mechanism. Consider the spherical fourbar mechanism (SFBM) shown in Figure 3.9 with links r1, r2, r3 and r4 respectively. The diagonal splits the mechanism into two spherical triangles with their sides being respectively r1, r2, and r3, r4, Note that r1, r2, r3, r4 and are all great arcs. The dihedral angles of the spherical triangles are represent ed with lower-case greek letters. Figure 3.9: Spherical Four-Bar Mechanism with Links r1, r2, r3 and r4. By applying the Law of Cosines to each of the two spherical triangles respectively, knowing the link parameters and 2 one can solve for all the dihedral angles and the common side,

PAGE 32

23Applying the spherical Law of Cosines to the spherical triangle on the left hand side in Figure 3.9, we get: 2 2 1 2 1 1cos sin sin cos cos cos r r r r (3.22) sin sin cos cos cos cos1 1 2 1r r r (3.23) sin sin cos cos cos cos2 2 1 1r r r (3.24) Applying the spherical Law of Cosines to the spherical triangle on the right hand side in Figure 3.9, we get: sin sin cos cos cos cos4 3 4 1r r r (3.25) sin sin cos cos cos cos4 4 3 1r r r (3.26) 4 3 4 3 1sin sin cos cos cos cos r r r r (3.27) Using the trigonometric ident ities developed in this chapter, it will be possible to analyze the kinematics of a s pherical mechanism in the next chapter.

PAGE 33

24Chapter 4 Bistability of a Spherical Four-Bar Mechanism This chapter provides an overview on how the principle of virtual work is used to determine bistability. The positi on and energy equations of a SFBM are then developed. 4.1 Principle of Virtual Work To determine the bistability of a s pherical compliant mechanism, one way is to apply the principle of virtual work. “The net virtual work of all active forces is zero if and only if an ideal mechanical sy stem is in equilibrium.” In the case where the energy equations and position equati ons of a mechanism are available the following procedure applies [15]. Position analysis. Obtain the position equations for the mechanism. Energy equations. Develop the equations that express the energy stored in the springs of the pseudo-rigid body model of the mechanism. First derivative. Take the first derivative of the energy equations with respect to the generalized coor dinate. The resulting equation corresponds to the virtual work equat ion of the mechanism. It gives

PAGE 34

25 the relationship between an appli ed displacement and the reaction moment. Equilibrium positions. Solve for all values of the generalized coordinate for which the first derivative of the energy equation is zero. These correspond to t he equilibrium positions. Stable positions. Differentiate the energy equ ation again to find the second derivative of the energy equations with respect to the generalized coordinate. The sign of the result will determine if the equilibrium position is stable or unstable. 4.2 Virtual Work Equations of a Compliant Spherical Four-Bar Mechanism Consider the spherical mechanism of Fi gure 3.7 but with four small-lengthflexural-pivots replacing the four revolute joints as shown in Figure 4.1-a). The position equations are already develop ed, see equations 3. 22-3.27.The energy equation, similarly to a planar mechani sm, can be easily developed using the pseudo-rigid-body model of t he compliant spherical four-bar mechanism (CSM) of Figure 4.1-b). The model has four links r1, r2, r3, r4 and four torsional springs; acting as four small length flexural pivots; with constants being k1, k2, k3 and k4. To determine bistability, each possible torsional spring may be examined independently of the others, by choosing its constant, ki, to be non-zero while setting the other springs’ constants equal to zero. Thus a procedure is developed

PAGE 35

26to determine the bistable behavior due to each spring. The same procedure can then be applied to the other remaining springs. a) b) Figure 4.1: a) A Compliant Spherical Fou r-Bar Mechanism and Its b) PRBM

PAGE 36

27Recall from Section 3.1.2.2 that a conv enient form of virtual work is found from the derivative of the potential energy, V with respect to the generalized coordinate q q dq dV W (4.1) Also, the virtual work W due to a moment input,inM, and a virtual displacement, is M W (4.2) The potential energy stored in any of the four springs is: 22 1initial finalk V (4.3) Where initial is the unstressed orientati on of the torsional spring. As mentioned before, stabl e equilibrium positions can obtained by taking the first derivative of the energy equat ions with respect to the generalized coordinate and then solving for all values of the generalized coordinate for which the input torque required to maintain a position is zero.

PAGE 37

28Choosing 2 as the generalized coordinate and differentiating with respect to2 the generalized virtual work equat ion for each torsional spring is: 2 2 2 2 2 2 in initial final i i i inM d d k M d dV W (4.4) inM is the input torque due to a force applied to link r2. The displacement coordinate associated with inM is the input rotation, 2 At equilibrium, the virtual work done W by the torsional springs is assumed balanced by the work done by the input torque. Rearranging equation 4.4: 2 , d d k Minitial i final i i i in (4.5-a) Equilibrium positions for a particular torsional spring 2 ,0 d d k Minitial i final i i in (4.5-b) The first part of Equation 4.5-b, initial i final i ik, , describes the linear part of the moment due to the torsional spring. It pr ovides only one solution to equation 4.5b, initial i final i , ,. The second part of Equation 4.5-b, 2 d d,the kinematic coefficient, can be non linear and may result in other equilibrium positions when it is equal to zero. This procedure can be used to eval uate the input moment required at each of the four joints.

PAGE 38

294.3 A Simplified Mathematical Model of a Bistable CSM In the preceding section, a procedure for determining bistability in a general spherical compliant four-bar mechanism was developed. In this section, a specific four-bar-mechanism c onfiguration is considered. Figure 4.2: PRBM of a Spherical Mechanism with Known Link Lengths The PRBM of a spherical mechan ism which has great arcs as links, with angular measures of r1=30, r2=90 r3=90, r4=60 is presented in Figure 4.2. Using these specific angular measurem ents instead of arc length allows for derivations that are independent of the radius of the particular sphere and simplifies the process for determining bistability.

PAGE 39

30Recalling equations 4.5-a and 4.5-b with 2 12 23 34 41 An examination of the moment-rotation cu rve for each joint shows which of the four springs produces two stable positi ons within the allowable motion of the mechanism. 4.3.1 Joint 1: The Flexura l Pivot Connecting Links r1 and r2 Using equation 4.5, the moment input required to bend “joint 1” becomes For 2 12 i 2 12 12 12 12 d d k Minitial final in (4.6) Where initial 12 is the undeflected position of the small length flexural pivot and 12 12 d d. Because 12 12 d d, the moment-rotation curve is linear. Thus, the elasticity associated with “j oint 1” does not produce bi stable behavior. One stable position can be seen from that graph and it is located at point A of zero rotation of 2

PAGE 40

31 Figure 4.3: Input Moment Required to Hold the Spherical Four-Bar Mechanism in Equilibrium for Joint 1 4.3.2 Joint 2: The Flexural Pivot Connecting Links r2 and r3 Using equation 4.6 the moment equation is: For 23 i 2 34 23 23 23 ,) ( d d k Minitial final in (4.7) Where 2 2 2 23 d d d d d d (4.8) A

PAGE 41

32Using equations 3.24 and 3.25 yields: 2 3 3 4 1 2sinsin coscoscos cos d r rr d d d (4.9) And 2 2 2 1 1 2sinsin coscoscos cos d r rr d d d (4.10) Since r2=90 and r3=90 in the case of our mechani sm, simplification yields to: 2 3 4 1 2sinsin cos cos d r r d d d (4.11) And 2 2 1 1 2sinsin cos cos d r r d d d (4.12) Differentiating equations 4.11 and 4.12 and substituting back in equation 4.8 yields to: sin cossin sin sinsincos1 1 2 23r r d d (4.13)

PAGE 42

33 Figure 4.4: Input Moment Required to Hold the Spherical Four-Bar Mechanism in Equilibrium for Joint 2 The moment-rotation curve of Figure 4.4 shows that when joint 2 is the only compliant joint, the mechanism has on ly two equilibrium positions. The first would be when no input rotation is applied, represented by point A in Figure 4.4, and it is a position of stabl e equilibrium. The second position, a position of unstable equilibrium where the slope of the moment input curve is negative, marked by point B in Figure 4.4. A B

PAGE 43

344.3.2 Joint 3: The Flexural Pivot Connecting Links r3 and r4 Using equation 4.6 the moment a joint 3 is: For 34i 2 34 34 34 34 ,) (0 d d k Min (4.14) Using equation 3.27: 2 4 3 4 3 1 2 2 34sin sin cos cos cos cos d r r r r d d d d d (4.15) Taking the derivative yields: 4 3 2 2 2 2 1 2 34sin sin 1 1 sin sin sin sinr r V U r r d d (4.16) Where U : 4 3 4 3sin sin cos cos cosr r r r U (4.17) and V : 2 2 1 2 1cos sin sin cos cos r r r r V (4.18) Substitution of U and V in 4.16 and simplifying yields to 4 3 2 2 1 2 34sin sin sin sin sin sinr r r r d d (4.19) It can be concluded by examining t he moment –rotation curve seen in Figure 4.5 that elasticity associated with joint 3 produces bistable behavior. At 0 rotation, represented by point A on the graph of Figure 4.5, the mechanism is in

PAGE 44

35its first stable equilibrium position. When the input link r2 is rotated by 90o, the mechanism reaches its non-stable equilib rium position represent by point B on the curve of Figure 4.5. The mechanism reaches its second stable equilibrium position, when the input link r2 is rotated by 180o to point C on the graph. Furthermore, the maximum torque levels reached are at points D and E Figure 4.5: Input Moment Required to Hold t he Spherical Four-Bar Bechanism in Equilibrium for Joint 3 A B C D E

PAGE 45

364.3.4 Joint 4: The Flexural Pivot Connecting Links r4 and r1 Using equation 3.4 the input moment is: For 34i 2 41 41 41 41 ,) ( d d k Minitial final in (4.14) Where 2 2 2 41 d d d d d d (4.15) Using equations 3.23 and 3.26 yields: 2 3 4 3 1 2sin sin cos cos cos cos d r r r d d d (4.16) And 2 1 1 2 1 2sin sin cos cos cos cos d r r r d d d (4.17) Since r2=90 and r3=90in the case of our mechanism, simplification yields: 2 3 4 1 2sin sin cos cos cos d r r d d d (4.18) And 2 1 1 1 2sin sin cos cos cos d r r d d d (4.19)

PAGE 46

37Now differentiating equations 4.18 and 4.19 yields: sin sin sin sin sin csc cot2 2 1 2 4 2 r r r d d (4.20) And sin sin sin sin sin cot2 2 1 1 2 r r r d d (4.21) Simplifying equations 4.20 and 4. 21 and substituting back gives: 2 1 1 4 1 4 2 41sin sin cos sin sin sin sin sin cos r r r r r d d (4.22) Figure 4.6: Input Moment Required to Hold the Spherical Four-Bar Mechanism in Equilibrium for Joint 4

PAGE 47

38 The moment-rotation curve of Figure 4.6 shows that the elasticity associated with joint 4 produces only one stable pos ition located at the 0 point. After examining each joint independently it can be determined that the only compliant joint that allows the mechanism to be bistable is joint 3. As for the other joints, they can be compli ant but small enough that their k constant is very low relatively to k3 of joint 3. The total moment-rotati on would then have the similar non-linear shape as the curve of Figure 4.5. In this chapter, it was demonstrated ma thematically that for small length flexural pivots, bistability in a spherical compliant mec hanism is possible. In the next chapter, the design and analysis of a bistable spherical co mpliant four-bar micromechanism is described based on t he analysis presented in this chapter.

PAGE 48

39Chapter 5 A Bistable Spherical Co mpliant Micromechanism Micromechanisms with accurate out-off-plane motion and low power consumption are needed and might be useful in several applications [6]. One possible method to achieving that goal is to design a bistable, compliant, spherical mechanism (BSCM). However, when dealing with mi cromachining, the design and manufacturing of a bistable, compliant, spherical micromechanism is somewhat challenging because the device is fabricated in plane but its motion is intended to be out-of-plane. In this chapter, the des ign, fabrication and analysis of a BSCM based on the model developed in Section 4.2 will be presented. 5.1 Fabrication Process The design of the BSCM followed the design rules set by the micromachining process chosen for fabricat ion; the Multi-User MEMS Processes (MUMPs). MUMPs is a threelayer polysilicon surface micromachining process. Figure 5.1 shows a cross section of the three-layer polysilicon surface micromachining. This proc ess has the general featur es of a standard surface micromachining process: (1) polysilicon is used as the structural material, (2) deposited oxide (PSG) is used as the sa crificial layer and silicon nitride

PAGE 49

40is used as electrical isolation betw een the polysilicon and t he substrate [30]. Figure 5.1: Cross Sectional View Showing a ll 7 Layers of the MUMPs Process [30] Tables 5.1 lists the material name, th ickness and lithography level name of each layer in MUMPS. Table 5.2 shows the minimum feature size for each corresponding layer. Table 5.1: Layer Names, Thicknesses and Lithography Levels [30] Metal Metal

PAGE 50

41Table 5.2: Minimum Feature Size per Lay er of Polysilicon [30] 5.2 Design The BSCM was designed us ing computer aided design (CAD) software LEdit, developed by Tanner Tools. Figure 5. 2 represents a simplified sketch of the BSCM. Two different models were designed: Model 1 where joint 2 was made compliant and another Model 2 with joint 2 was a revolute pin joint. After preliminary testing, Model 1 was chosen because it was more reliable and easier to design. Figure 5.3 is a scanning el ectron microscope (SEM) image of the Model 1 BSCM in its fabricated position. The BSCM has three basic components: Two sliders and a spherical-f our bar mechanism with links r1, r2, r3, and r4 seen in Figures 5.2 and 5.3. r1 is the ground link, r2 the input link, r3 the coupler link and r4 the follower link. Links r2 and r4 are joined to the subs trate by a staple hinge that allows 180 rotation. Link r3 is connected to r2 and r4 by compliant joints as

PAGE 51

42shown in Figure 5.2. The axes of rotation of the four joints intersect at a single point. The sliders act as mechanical actuators and are connected each to the input link r2, by staple hinges. The mechanism in its fabricated position is shown in Figure 5.3 which is its first stabl e equilibrium positio n. By moving the Raising Slider to the left, link r2 will rotate and links r3 and r4 will move out-of-plane as shown in Figure 5.5. In order to bring the mechanism back to its original position, Lowering Slider would be moved to the right. Figure 5.2: BSCM Showing the Nomencla ture for the Mechanism

PAGE 52

43 Figure 5.3: Scanning Electron Microscope (SEM) Image of BSCM as Fabricated: First Stable Equilibrium Position Figure 5.4: SEM Image of the Staple Hinge

PAGE 53

44 Figure 5.5: SEM Image of the BSCM in an Out-Of-Plane Position 5.3 BSCM Testing In order to test several configurat ions of the mechanism, eight models were designed and fabricated. Each model differed from t he others in the length of the joint labeled A is Figure 5.5, this changes the stiffness k of that joint thus affecting the location of the stable equilibrium positions. In order to differentiate the different models on the polysilicon die, each mechanism configuration was marked with a number of squares embedded in the substrate. Testing was done under an optical microscope fitted with a probing station. Two micro-probes with three degrees of freedom (DOF) were pl aced on the right and left hand side of the prototype. During test ing the probes were used to push/pull the slider thus actuating the BSCM. A

PAGE 54

45As mentioned before the BSCM is in its first stable equilibrium position when it is in its “as fabricated position” seen in Figure 5.3. Figure 5.6: SEM Images of BSCM in its Se cond Stable Equilibrium Position Figure 5.6 shows the BSCM in its se cond stable equilibrium position. In this second stable position, the mechanism is in a position in which most of its links are no longer in cont act with the substrate (or ground plane). The stability of this second position has been demons trated experimentally by probing the device at various points along its lengt h and its demonstrating resilience to loading (i.e. always returning to the second stable position). 5.4 Finite-Element Analysis The BSCM prototype differed in an important way from the elastic mathematical models descri bed in Chapter 4 in that the entire links were more flexible, thus deflections occurred along th e length of the mechanism’s links and not just at the short length flexural pi vots. The increased flexibility resulted in important qualitative differ ences in the mechanism’s stability behavior and its

PAGE 55

46input angle-input torque relation. A Finite -Element Analysis (FEA) (using ANSYS) of the input torque required to hold t he mechanism at equilibrium at a given rotation is shown in Figure 5.8. Figure 5.7: Mathematical Model of the Mo ment-Rotation Relationship for a BSCM with Rigid Links and Short-Length Flexural Pivots Figure 5.8: Moment-Rotation Relationship fr om a FEA of the BSCM Prototype

PAGE 56

47 The moment-rotation curve shown in Figure 5.8 shows qualitative differences compared to the curve shown in Figure 5.7. In its initial position, the spherical bistable mechanism is at its first stable equilibr ium position, as seen in the mechanism configuration shown in Fi gure 5.3 and is represented by point A in Figure 5.8. The input link is then rotated past 140 degrees to its maximum torque level at point B Then, due to nonlinear deflections in the compliant mechanism, the torque begins to drop off. Intriguingly, both te sting and analysis show the existence of a point C past which further rotation of the input link is impossible. At this point, releasing the input causes it to move to a second stable position at D rather than back to it s original position at A Furthermore, because of the rotation limit at point C it proves very difficult if not impossible to cause the mechanism to return to its original positi on by rotating the input link. Auxiliary actuation at the second ground point s eems to be required to get the mechanism to return to its original configurat ion. Thus, the region between points C and E are elastically locked in that the me chanism cannot spontaneously leave this region without auxiliary actuation. Fini te Element models show that even very large torques do not cause the mechanism at angle C to rotate further. This suggests that the mechanism has potential fo r high structural strength at or near that position.

PAGE 57

48Chapter 6 6.1 Conclusions This paper has discussed t he design of an innovative device: A bistable spherical compliant fourbar mechanism. This device offers many valuable features, such as: Two stable positions that require power only when moving from one stable position to the other, pr ecise and repeatable out of plane motion with resistance to small perturbations. The equations for position and input torque have been obtained. The device was fabricated using the MUMPs surface micromachining process. Bistabilty wa s demonstrated through testing done on a micro-prototype. Compared to the PRBM with small length flexural pivots of the BSCM, Finite-element models of the BSCM indicated im portant qualitative difference in the mechanism’s stabilit y behavior and its input-angle-input moment relation and that may be due to the defle ction of the ‘rigid ’ members of the mechanism. 6.2 Recommendations for Future Works Results from FEA call fo r further studies on the BSCM prototype. Stress and force analysis would be helpful in optimizing the BSCM.

PAGE 58

49 Based on the theory developed in this thesis, prototypes with different parameters could be manufactured and test ed for more accurate results. In this research, actuation of the BSCM was perform ed using mechanical micro-probes. Further act uation methods, i.e. The rmomechanical In plane Microactuators (TIM) shoul d be studied and tested. Micromachining cost increases with t he increase in the number of layers required to build a microprotytpe. For cheaper micromachining, designing a single layer fully compliant BSCM would be helpful.

PAGE 59

50References [1] K. J. Gabriel, “Microelectr omechanical Systems (MEMS) Tutorial ,” IEEE Test Conference (TC), pp. 432-441, 1998. [2] M. Mehregany and M. Huff, “Microelectromec hanical Systems," Proceedings of the IEEE Cornell C onference on Advanced Concepts in High Speed Semiconductor Device s and Circuits, pp. 9-18, 1995. [3] C. Lusk, “Ortho-Planar Mechanisms for Microelctromechanical Systems,” Dissertation, Brigham Young University, Provo, UT. [4] Lam, A.H.F.; Li, W.J.; Yunhui Liu; Ning Xi, "MIDS: micro input devices system using MEMS sensors," Intelligent Robots and System, 2002. [5] Fukushige, T.; Hata, S.; Shimokohbe, A., "A MEMS conical spring actuator array," Microelectromechanical Systems, Journal of vol.14, no.2, pp. 243253, April 2005. [6] R. S. Payne, “MEMS commerci alization: Ingredi ents for success," Proceedings of the IEEE Micro Electr o Mechanical Systems (MEMS), pp. 7-10, 2000. [7] Texas Instruments, “Digit al Light Processors,” http://www.dlp.com/tech/what.aspx 2007. [8] J. A. Bradley, “Design of Surf ace Micromachined Compliant MEMS,” Thesis, Iowa State University, Ames, IA. [9] J. Parise, L. Howell, and S. Magleby, “Ortho-planar mechanisms," Proceedings of the 2000 ASME De sign Engineering Technical Conferences, DETC2000/MECH 14193, pp. 1-15, 2000. [10] Ananthasuresh, GK and Howell, Larry L (2005) “Mechanical Design of Compliant Microsystems-A Pe rspective and Prospects”. Journal of Mechanical Design 127(4):pp. 736-738. [11] Felton, B. “Better robots th rough clean living”. Intec, May 2001.

PAGE 60

51[12] R. Cragun and L. L. Howell, “A New Cons trained Thermal Expansion Micro-Actuator," American Society of Mechani cal Engineers, Dynamic Systems and Control Division (Publication) DSC vol. 66, pp. 365-371, 1998. [13] C. D. Lott, J. Harb T. W. McClain, and L. Howe ll, “Dynamic Modelling of a Surface-Micromachined Linear Thermomechanical Actuator," Technical Proceedings of the Four th International Conf erence on Modeling and Simulation of Microsystems, MSM 2001, Hilton Head Island, South Carolina, pp. 374-377, 2001. [14] Howell, L.L. and Midha, A., 1994, "A Method for the Design of Compliant Mechanisms with Small-Length Flexural Pivots," ASME Journal of Mechanical Design, Vol. 116, No. 1, pp. 280-290. [15] L. Howell, Compliant Mechanisms. New York: Wiley-Interscience, 2001 [16] Howell, L.L., Rao, S.S., and Mi dha, A., 1994, "The Reliability-Based Optimal Design of a Bistable Compliant Mechanism," ASME Journal of Mechanical Design, Vol. 116, No.4, pp. 1115-1121. [17] Jensen, B.D., Howell, L.L., Gunyan, D.B., and Salmon, L.G., 1997, "The Design and Analysis of Compliant MEMS Using the Pseudo-Rigid-Body Model," Microelectomechanical Systems (MEMS) 1997, presented at the 1997 ASME International Mechanical Engineering Congress and Exposition, November 1621, 1997, Dallas, Texas, DSC-Vol. 62, pp. 119126. [18] Jensen, B.D., Howell, L.L., and Salmon, L.G., 1998, "Introduction of TwoLink, In-Plane Bistable Compliant ME MS," Proceedings of the 1998 ASME Design Engineering Technical Conferences, DETC98/MECH-5837. [19] Jensen, B.D., "Identification of Macroand Micro-Compliant Mechanism Configurations Resulting in Bistabl e Behavior," M.S. Thesis, Brigham Young University, Provo, Utah. [20] Jensen, B.D., Howell, L.L., Gunyan, D.B., and Salmon, L.G., 1997, "The Design and Analysis of Compliant MEMS Using the Pseudo-Rigid-Body Model," Microelectomechanical Systems (MEMS) 1997, presented at the 1997 ASME International Mechanical Engineering Congress and Exposition, November 1621, 1997, Dallas, Texas, DSC-Vol. 62, pp. 119126. [21] SHIGLEY, J.E. WICKER, J.J. Theory of machines & mechanisms McGraw-Hill, 1980.

PAGE 61

52[22] B.A. Coulter and R.E. Miller, Numerical analysis of a generalized plane elastica with non-linear material behavior, Int J Numer Meth Eng 26 1988, pp. 617–630. [23] Howell, L.L. and Midha, A., 1994, "A Method for the Design of Compliant Mechanisms with Small-Length Flexural Pivots," ASME Journal of Mechanical Design, Vol. 116, No. 1, pp. 280-290. [24] Howell, L.L., and Midha, A., 1996, "A Loop-Closure Theory for the Analysis and Synthesis of Compliant Mechanisms," ASME Journal of Mechanical Design, Vol. 118, No. 1, pp. 121-125. [25] Salmon, L.G., Gunyan, D.B., Der derian, J.M., Opdahl, P.G., and Howell, L.L., 1996, "Use of the Pseudo-Rig id Body Model to Simplify the Description of Compliant Micro-Mechanisms," 1996 IEEE Solid-State and Actuator Workshop Hilton Head Island, SC, pp. 136-139. [26] Salamon, B.A., and Midha, A., 1992, "An Introduction to Mechanical Advantage in Compliant Mechanisms," Advances in Design Automation, (Ed.: D.A. Hoeltzel), DE-Vol 44-2, 18th ASME Design Automation Conference, pp. 47-51. [27] Howell, L.L., Midha, A ., and Norton, T.W., 1996, "E valuation of Equivalent Spring Stiffness for Use in a Pseudo-Rigid-Body Model of LargeDeflection Compliant Mechanisms," ASME Journal of Mechanical Design, Vol. 118, No. 1, pp. 126-131. [28] M. R. Spiegel and J. Liu, Schaum's Outlines: Mathematical Handbook of Formulas and Tables. New York, NY: McGraw-Hill, 1999. [29] D. W. Henderson, Experiencing G eometry: In Euclidean, Spherical, and Hyperbolic Spaces, 2nd Edition. Upper Saddle River, NJ: Prentice Hall, 2001. [30] D. Koester, R. Mahadevan, B. Hardy, and K. Markus, MUMPs Design Handbook.Research Triangle Park, NC: Cronos Integrated Microsystems, 2001.

PAGE 62

53 Appendices

PAGE 63

54Appendix A: ANSYS Batch Files The following files are test files wri tten in notepad. They are Batch files used by ANSYS to run FEA and determi ne the moment input on the microprototype. Each file contains different prototype dimensions. Batch File 1: !******************** **************** /CONFIG,NRES,1000000 !/CWD,'C:\Documents and Settings\aleon2\Desktop\Work' !******************** **************** !******************** ******************** !******* Set Up Model Variables ********* !******************** ******************** !*DO,asp, .1,.7,.3 !asp =.1 !aspect = 10*asp !*DO,arclength,1,120,1 !arclength=10 /title,3D Beam Non-linear Deflection /PREP7 !LCLEAR, ALL !LDELE, ALL !KDELE, ALL R=313.38 length in micrometers PI=acos(-1.) h1=2 b1=20

PAGE 64

55Appendix A (Continued) b2=5 h2=2 b3=27.6 h3=2 !*********** Area prope rties ************** A1 = h1*b1 Iy1= 1/12*b1*h1*h1*h1 Iz1= 1/12*h1*b1*b1*b1 E1= 169E3 Young's modulus in MPa, Force will be micro Newtons !******************** **************** A2= h2*b2 Iz2= 1/12*h2*b2*b2*b2 Iy2= 1/12*b2*h2*h2*h2 E2= 169e3 !******************** **************** A3= h3*b3 Iz3= 1/12*h3*b3*b3*b3 Iy3= 1/12*b3*h3*h3*h3 E3= 169e3 !********** Declare an element type: Beam 4 (3D Elastic) ********* ET,1,BEAM4 KEYOPT,1,2,1 KEYOPT,1,6,1

PAGE 65

56Appendix A (Continued) R,1,A1,Iy1,Iz1,h1,b1, !******Che ck on the assumptions being made ****** R,2,A2,Iy2,Iz2,h2,b2, R,3,A3,Iy3,Iz3,h3,b3, MPTEMP,1,0 MPDATA,EX,1,,E1 MPDATA,PRXY,1,,0.35 Material proper ties for material 1 and 2 MPTEMP,1,0 MPDATA,EX,2,,E2 MPDATA,PRXY,2,,0.35 MPTEMP,1,0 MPDATA,EX,3,,E3 MPDATA,PRXY,3,,0.35 !******************** **************** !********** Create Keypoints 1 throug 7: K(Point #, X-Co ord, Y-Coord, Z-Coord) ********* K,1,0,0,0, K,2,0,-225,0, K,3,0,-50,0, K,4,0,50,0, K,5,R*cos(PI/180*72),R*SIN(PI/180*72),0, K,6,R*cos(PI/180*33),R*sin(PI/180*33),0, K,7,R*cos(PI/180*29),R*sin(PI/180*29),0, K,8,R*cos(PI/180*44),R*sin(PI/180*44),0, !********* Create Beam using Lines and Ar cs and divide into segments ********* LSTR, 2, 3 LSTR, 4, 5 LSTR, 3, 4 Draws lines connec ting keypoints 1 through 7 LSTR, 6, 7

PAGE 66

57Appendix A (Continued) LARC,5,6,1,R, Defines a circular arc LARC,7,8,1,R, LESIZE,ALL,,,15 Specifies the divisions and spacing ratio on unmeshed lines, *****Try making 32 smaller !*********** MESH *********** real,3 Use real constant set 3 type,1 Use element type 1 mat,3 use material property set 3 LMESH,1,2 mesh lines 1-2 real,2 Use real constant set 2 type,1 Use element type 1 mat,2 use material property set 2 LMESH,3,4 mesh line 3-4 real,1 Use real constant set 1 type,1 Use element type 1 mat,1 use material property set 1 LMESH,5,6 mesh line 5-6 !******* Get Node Numbers at chosen keypoints ******* ksel,s,kp,,2 nslk,s *get,nkp2,node,0,num,max Retrieves a va lue 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

PAGE 67

58Appendix A (Continued) *get,nkp4,node,0,num,max nsel,all ksel,all ksel,s,kp,,5 nslk,s *get,nkp5,node,0,num,max nsel,all ksel,all ksel,s,kp,,6 nslk,s *get,nkp6,node,0,num,max nsel,all ksel,all ksel,s,kp,,7 nslk,s *get,nkp7,node,0,num,max nsel,all ksel,all ksel,s,kp,,8 nslk,s *get,nkp8,node,0,num,max nsel,all ksel,all FINISH !********************************* ************************** !********************** SOLUTION ************** ************* !********************************* ************************** /SOL ANTYPE,0 Specifies 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 e ffects in a static or full transient analysis

PAGE 68

59Appendix A (Continued) !AUTOTS, ON !CNVTOL,U,,0.000001,,0 !CNVTOL,F,,0.0001,,0 Sets c onvergence values for nonlinear analyses !************************* *********** DK,2, ,0, , ,UX,UY,UZ, ROTX,ROTZ, Boundary co nditions on keypoint 2 DK,3, ,0, , ,UX,UY,UZ, ROTX,ROTZ, Boundary co nditions on keypoint 2 DK,8, ,0, , ,UX,UY,UZ,ROTZ, LOCAL,11,CART,0,0,0,-44,0,0, CSYS,11 DK,8,ROTY,0 CSYS,0 !******************** **************** *DIM,my1,TABLE,10000 lsnum =0 *DO,step,1,120,1 theta=-1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO stop1 = lsnum !Arclen,On !*DO,step,143,146,.01 !theta = -1*step !DK,2,ROTY,theta*PI/180 !lsnum=lsnum+1 !LSWRITE,lsnum !*ENDDO DKDELE,2,ROTY

PAGE 69

60Appendix A (Continued) *DO,step,-20750,33250,250 !Maximum Moment Value 2.1065e+004 mmy1 = step*1 FK,2,MY,mmy1 lsnum=lsnum+1 *set,MY1(lsnum),step*1 LSWRITE,lsnum *ENDDO FKDELE,2,my *DO,step,-56,-22,1 theta=1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO DKDELE,2,ROTY *DO,step,36500,37725,25 !M aximum Moment Value 2.1065e+004 mmy1 = step*1 FK,2,MY,mmy1 lsnum=lsnum+1 *set,MY1(lsnum),step*1 LSWRITE,lsnum *ENDDO FKDELE,2,my *DO,step,-23.3,-24,-.1 theta=1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO

PAGE 70

61Appendix A (Continued) *DO,step,-24,-30,-1 theta=1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO *DO,step,-30,-180,-1 theta=1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO *Do,nn,1,lsnum LSSOLVE,nn /output,progress,txt,,append *VWRITE,nn Writes data to a file in a formatted sequence %16.8G /output *enddo /STATUS,SOLU FINISH !******************** ********* !******* GET RESULTS ********* !******************** ********* loadSteps=lsnum /POST1 *DIM,rotY2,TABLE,loadSteps *DIM,disX3,TABLE,loadSteps *DIM,disY3,TABLE,loadSteps *DIM,disZ3,TABLE,loadSteps *DIM,disX5,TABLE,loadSteps *DIM,disY5,TABLE,loadSteps *DIM,disZ5,TABLE,loadSteps

PAGE 71

62Appendix A (Continued) !*DIM,fx1,TABLE,loadSteps !*DIM,fy1,TABLE,loadSteps !*DIM,fz1,TABLE,loadSteps !*DIM,mx1,TABLE,loadSteps !*DIM,mz1,TABLE,loadSteps *DIM,momy2,TABLE,loadSteps !*DIM,fx2,TABLE,loadSteps !*DIM,fy2,TABLE,loadSteps !*DIM,fz2,TABLE,loadSteps !*DIM,mx2,TABLE,loadSteps !*DIM,my2,TABLE,loadSteps !*DIM,mz2,TABLE,loadSteps !*DIM,fx3,TABLE,loadSteps !*DIM,fy3,TABLE,loadSteps !*DIM,fz3,TABLE,loadSteps !*DIM,mx3,TABLE,loadSteps !*DIM,my3,TABLE,loadSteps !*DIM,mz3,TABLE,loadSteps !*DIM,fx5,TABLE,loadSteps !*DIM,fy5,TABLE,loadSteps !*DIM,fz5,TABLE,loadSteps !*DIM,mx5,TABLE,loadSteps !*DIM,my5,TABLE,loadSteps !*DIM,mz5,TABLE,loadSteps *Do,nn,1,lsnum set,nn *GET,roty,Node,nkp2,ROT,Y *SET,rotY2(nn),roty *GET,my2,Node,nkp2,RF,MY *SET,MOMY2(nn),my2 *ENDDO /output,output_arc%arclengt h%_asp%aspect%,txt,, *MSG,INFO,'t','w','R','E','arclength' Writes an output message via the ANSYS message subroutine %-8C %-8C %-8C %-8C %-8C

PAGE 72

63Appendix A (Continued) *VWRITE,h2,b2,R,E2,arclength Writes data to a file in a formatted sequence %16.8G %-16.8G %-16 .8G %-16.8G %-16.8G *MSG,INFO,'roty2','my1','my2' %-8C %-8C *VWRITE,rotY2(1),MY1(1),MOMY2(1) %16.8G %-16.8G %-16.8G /output FINISH !*ENDDO !*ENDDO Batch File 2 !******************** **************** /CONFIG,NRES,1000000 !/CWD,'C:\Documents and Settings\aleon2\Desktop\Work' !******************** **************** !******************** ******************** !******* Set Up Model Variables ********* !******************** ******************** !*DO,asp, .1,.7,.3 !asp =.1 !aspect = 10*asp !*DO,arclength,1,120,1 !arclength=10 /title,3D Beam Non-linear Deflection

PAGE 73

64Appendix A (Continued) /PREP7 !LCLEAR, ALL !LDELE, ALL !KDELE, ALL R=313.38 length in micrometers PI=acos(-1.) h1=2 b1=25 b2=5 h2=2 b3=27.6 h3=2 b4=3.5 h4=2 !*********** Area prope rties ************** A1 = h1*b1 Iy1= 1/12*b1*h1*h1*h1 Iz1= 1/12*h1*b1*b1*b1 E1= 169E3 Young's modulus in MPa, Force will be micro Newtons !******************** **************** A2= h2*b2 Iz2= 1/12*h2*b2*b2*b2 Iy2= 1/12*b2*h2*h2*h2 E2= 169e3 !******************** **************** A3= h3*b3

PAGE 74

65Appendix A (Continued) Iz3= 1/12*h3*b3*b3*b3 Iy3= 1/12*b3*h3*h3*h3 E3= 169e3 !******************** **************** A4= h4*b4 Iz4= 1/12*h4*b4*b4*b4 Iy4= 1/12*b4*h4*h4*h4 E4= 169e3 !******************** **************** !********** Declare an element type: Beam 4 (3D 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, !******Che ck on the assumptions being made ****** R,2,A2,Iy2,Iz2,h2,b2, R,3,A3,Iy3,Iz3,h3,b3, R,4,A4,Iy4,Iz4,h4,b4, MPTEMP,1,0 MPDATA,EX,1,,E1 MPDATA,PRXY,1,,0.35 Material proper ties for material 1 and 2 MPTEMP,1,0 MPDATA,EX,2,,E2 MPDATA,PRXY,2,,0.35 MPTEMP,1,0 MPDATA,EX,3,,E3

PAGE 75

66Appendix A (Continued) MPDATA,PRXY,3,,0.35 MPTEMP,1,0 MPDATA,EX,4,,E4 MPDATA,PRXY,4,,0.35 !******************** **************** !********** Create Keypoints 1 throug 7: K(Point #, X-Co ord, Y-Coord, Z-Coord) ********* K,1,0,0,0, K,2,0,-225,0, K,3,0,-50,0, K,4,0,50,0, K,5,R*cos(PI/180*72),R*SIN(PI/180*72),0, K,6,R*cos(PI/180*33),R*sin(PI/180*33),0, K,7,R*cos(PI/180*29),R*sin(PI/180*29),0, K,8,R*cos(PI/180*44),R*sin(PI/180*44),0, !********* Create Beam using Lines and Ar cs and divide into segments ********* LSTR, 2, 3 LSTR, 4, 5 LSTR, 3, 4 Draws lines connect ing keypoints 1 through 7 LSTR, 6, 7 LARC,5,6,1,R, Defines a circular arc LARC,7,8,1,R, LESIZE,ALL,,,15 Specifies the divisions and spacing ratio on unmeshed lines, *****Try making 32 smaller !*********** MESH *********** real,3 Use real constant set 3 type,1 Use element type 1 mat,3 use material property set 3 LMESH,1,2 mesh lines 1-2

PAGE 76

67Appendix A (Continued) type,1 Use element type 1 mat,2 use material property set 2 LMESH,4 mesh line 4 real,4 Use real constant set 4 LMESH,3 mesh line 3 real,1 Use real constant set 1 type,1 Use element type 1 mat,1 use material property set 1 LMESH,5,6 mesh line 5-6 !******* Get Node Numbers at chosen keypoints ******* ksel,s,kp,,2 nslk,s *get,nkp2,node,0,num,max Retrieves a va lue 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 nsel,all ksel,all ksel,s,kp,,5 nslk,s *get,nkp5,node,0,num,max nsel,all ksel,all

PAGE 77

68Appendix A (Continued) ksel,s,kp,,6 nslk,s *get,nkp6,node,0,num,max nsel,all ksel,all ksel,s,kp,,7 nslk,s *get,nkp7,node,0,num,max nsel,all ksel,all ksel,s,kp,,8 nslk,s *get,nkp8,node,0,num,max nsel,all ksel,all FINISH !********************************* ************************** !********************** SOLUTION ************** ************* !********************************* ************************** /SOL ANTYPE,0 Specifies the analysi s type and restart status and "0" means that it Performs a static analysis. Valid for all degrees of freedom NLGEOM,1 Includes large-deflection e ffects in a static or full transient analysis !AUTOTS, ON !CNVTOL,U,,0.000001,,0 !CNVTOL,F,,0.0001,,0 Sets c onvergence values for nonlinear analyses !************************* *********** DK,2, ,0, , ,UX,UY,UZ, ROTX,ROTZ, Boundary co nditions on keypoint 2 DK,3, ,0, , ,UX,UY,UZ, ROTX,ROTZ, Boundary co nditions on keypoint 2

PAGE 78

69Appendix A (Continued) DK,8, ,0, , ,UX,UY,UZ,ROTZ, LOCAL,11,CART,0,0,0,-44,0,0, CSYS,11 DK,8,ROTY,0 CSYS,0 !******************** **************** *DIM,my1,TABLE,10000 lsnum =0 *DO,step,1,215,15 theta=-1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO stop1 = lsnum DKDELE,2,ROTY *DO,step,-15100,15900,100 !Maximum Moment Value 2.1065e+004 mmy1 = step*1 FK,2,MY,mmy1 lsnum=lsnum+1 *set,MY1(lsnum),step*1 LSWRITE,lsnum *ENDDO FKDELE,2,my *DO,step,-56,-25,1 theta=1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO

PAGE 79

70Appendix A (Continued) DKDELE,2,ROTY *DO,step,12925,16975,25 !M aximum Moment Value 2.1065e+004 mmy1 = step*1 FK,2,MY,mmy1 lsnum=lsnum+1 *set,MY1(lsnum),step*1 LSWRITE,lsnum *ENDDO FKDELE,2,my *DO,step,-39,-180,-1 theta=1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO !*DO,step,-24,-30,-1 !theta=1*step !DK,2,ROTY,theta*PI/180 !lsnum=lsnum+1 !LSWRITE,lsnum !*ENDDO !*DO,step,-30,-180,-1 !theta=1*step !DK,2,ROTY,theta*PI/180 !lsnum=lsnum+1 !LSWRITE,lsnum !*ENDDO *Do,nn,1,lsnum LSSOLVE,nn /output,progress,txt,,append *VWRITE,nn Writes data to a file in a formatted sequence

PAGE 80

71Appendix A (Continued) %16.8G /output *enddo /STATUS,SOLU FINISH !******************** ********* !******* GET RESULTS ********* !******************** ********* loadSteps=lsnum /POST1 *DIM,rotY2,TABLE,loadSteps *DIM,disX3,TABLE,loadSteps *DIM,disY3,TABLE,loadSteps *DIM,disZ3,TABLE,loadSteps *DIM,disX5,TABLE,loadSteps *DIM,disY5,TABLE,loadSteps *DIM,disZ5,TABLE,loadSteps !*DIM,fx1,TABLE,loadSteps !*DIM,fy1,TABLE,loadSteps !*DIM,fz1,TABLE,loadSteps !*DIM,mx1,TABLE,loadSteps !*DIM,mz1,TABLE,loadSteps *DIM,momy2,TABLE,loadSteps !*DIM,fx2,TABLE,loadSteps !*DIM,fy2,TABLE,loadSteps !*DIM,fz2,TABLE,loadSteps !*DIM,mx2,TABLE,loadSteps !*DIM,my2,TABLE,loadSteps !*DIM,mz2,TABLE,loadSteps !*DIM,fx3,TABLE,loadSteps !*DIM,fy3,TABLE,loadSteps !*DIM,fz3,TABLE,loadSteps !*DIM,mx3,TABLE,loadSteps !*DIM,my3,TABLE,loadSteps !*DIM,mz3,TABLE,loadSteps

PAGE 81

72Appendix A (Continued) !*DIM,fx5,TABLE,loadSteps !*DIM,fy5,TABLE,loadSteps !*DIM,fz5,TABLE,loadSteps !*DIM,mx5,TABLE,loadSteps !*DIM,my5,TABLE,loadSteps !*DIM,mz5,TABLE,loadSteps *Do,nn,1,lsnum set,nn *GET,roty,Node,nkp2,ROT,Y *SET,rotY2(nn),roty *GET,my2,Node,nkp2,RF,MY *SET,MOMY2(nn),my2 *ENDDO /output,output_arc%arclengt h%_asp%aspect%,txt,, *MSG,INFO,'t','w','R','E','arcl ength' Wr ites an output message via the ANSYS message subroutine %-8C %-8C %-8C %-8C %-8C *VWRITE,h2,b2,R,E2,arclength Writes data to a file in a formatted sequence %16.8G %-16.8G %-16 .8G %-16.8G %-16.8G *MSG,INFO,'roty2','my1','my2' %-8C %-8C *VWRITE,rotY2(1),MY1(1),MOMY2(1) %16.8G %-16.8G %-16.8G /output FINISH !*ENDDO !*ENDDO

PAGE 82

73Appendix A (Continued) Batch File 3 !******************** **************** /CONFIG,NRES,1000000 !/CWD,'C:\Documents and Settings\aleon2\Desktop\Work' !******************** **************** !******************** ******************** !******* Set Up Model Variables ********* !******************** ******************** !*DO,asp, .1,.7,.3 !asp =.1 !aspect = 10*asp !*DO,arclength,1,120,1 !arclength=10 /title,3D Beam Non-linear Deflection /PREP7 !LCLEAR, ALL !LDELE, ALL !KDELE, ALL R=313.38 length in micrometers PI=acos(-1.) h1=2 b1=20 b2=5 h2=2 b3=27.6 h3=2 !*********** Area prope rties **************

PAGE 83

74Appendix A (Continued) A1 = h1*b1 Iy1= 1/12*b1*h1*h1*h1 Iz1= 1/12*h1*b1*b1*b1 E1= 169E3 Young's modulus in MPa, Force will be micro Newtons !******************** **************** A2= h2*b2 Iz2= 1/12*h2*b2*b2*b2 Iy2= 1/12*b2*h2*h2*h2 E2= 169e3 !******************** **************** A3= h3*b3 Iz3= 1/12*h3*b3*b3*b3 Iy3= 1/12*b3*h3*h3*h3 E3= 169e3 !********** 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, !******Che ck on the assumptions being made ****** R,2,A2,Iy2,Iz2,h2,b2, R,3,A3,Iy3,Iz3,h3,b3, MPTEMP,1,0

PAGE 84

75Appendix A (Continued) MPDATA,EX,1,,E1 MPDATA,PRXY,1,,0.35 Material proper ties for material 1 and 2 MPTEMP,1,0 MPDATA,EX,2,,E2 MPDATA,PRXY,2,,0.35 MPTEMP,1,0 MPDATA,EX,3,,E3 MPDATA,PRXY,3,,0.35 !******************** **************** !********** Create Keypoints 1 throug 7: K(Point #, X-Co ord, Y-Coord, Z-Coord) ********* K,1,0,0,0, K,2,0,-235,0, K,3,0,-60,0, K,4,0,60,0, K,5,R*cos(PI/180*72),R*SIN(PI/180*72),0, K,6,R*cos(PI/180*33),R*sin(PI/180*33),0, K,7,R*cos(PI/180*29),R*sin(PI/180*29),0, K,8,R*cos(PI/180*44),R*sin(PI/180*44),0, !********* Create Beam using Lines and Ar cs and divide into segments ********* LSTR, 2, 3 LSTR, 4, 5 LSTR, 3, 4 Draws lines connect ing keypoints 1 through 7 LSTR, 6, 7 LARC,5,6,1,R, Defines a circular arc LARC,7,8,1,R, LESIZE,ALL,,,15 Specifies the divisions and spacing ratio on unmeshed lines, *****Try making 32 smaller

PAGE 85

76Appendix A (Continued) !*********** MESH *********** real,3 Use real constant set 3 type,1 Use element type 1 mat,3 use material property set 3 LMESH,1,2 mesh lines 1-2 real,2 Use real constant set 2 type,1 Use element type 1 mat,2 use material property set 2 LMESH,3,4 mesh line 3-4 real,1 Use real constant set 1 type,1 Use element type 1 mat,1 use material property set 1 LMESH,5,6 mesh line 5-6 !******* Get Node Numbers at chosen keypoints ******* ksel,s,kp,,2 nslk,s *get,nkp2,node,0,num,max Retrieves a va lue 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 nsel,all ksel,all ksel,s,kp,,5

PAGE 86

77Appendix A (Continued) nslk,s *get,nkp5,node,0,num,max nsel,all ksel,all ksel,s,kp,,6 nslk,s *get,nkp6,node,0,num,max nsel,all ksel,all ksel,s,kp,,7 nslk,s *get,nkp7,node,0,num,max nsel,all ksel,all ksel,s,kp,,8 nslk,s *get,nkp8,node,0,num,max nsel,all ksel,all FINISH !********************************* ************************** !********************** SOLUTION ************** ************* !********************************* ************************** /SOL ANTYPE,0 Specifies the analysi s type and restart status and "0" means that it Performs a static analysis. Valid for all degrees of freedom NLGEOM,1 Includes large-deflection e ffects in a static or full transient analysis !AUTOTS, ON !CNVTOL,U,,0.000001,,0

PAGE 87

78Appendix A (Continued) !CNVTOL,F,,0.0001,,0 Sets c onvergence values for nonlinear analyses !************************* *********** DK,2, ,0, , ,UX,UY,UZ, ROTX,ROTZ, Boundary co nditions on keypoint 2 DK,3, ,0, , ,UX,UY,UZ, ROTX,ROTZ, Boundary co nditions on keypoint 2 DK,8, ,0, , ,UX,UY,UZ,ROTZ, LOCAL,11,CART,0,0,0,-44,0,0, CSYS,11 DK,8,ROTY,0 CSYS,0 !******************** **************** *DIM,my1,TABLE,10000 lsnum =0 *DO,step,1,149,1 theta=-1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO stop1 = lsnum DKDELE,2,ROTY *DO,step,-9400,30200,200 mmy1 = step*1 FK,2,MY,mmy1 lsnum=lsnum+1 *set,MY1(lsnum),step*1 LSWRITE,lsnum *ENDDO *DO,step,30200,30500,100

PAGE 88

79Appendix A (Continued) mmy1 = step*1 FK,2,MY,mmy1 lsnum=lsnum+1 *set,MY1(lsnum),step*1 LSWRITE,lsnum *ENDDO FKDELE,2,my *DO,step,-58,-24,1 theta=1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO DKDELE,2,ROTY *DO,step,31350,33075,25 !M aximum Moment Value 2.1065e+004 mmy1 = step*1 FK,2,MY,mmy1 lsnum=lsnum+1 *set,MY1(lsnum),step*1 LSWRITE,lsnum *ENDDO FKDELE,2,my *DO,step,-28,-180,-1 theta=1*step DK,2,ROTY,theta*PI/180 lsnum=lsnum+1 LSWRITE,lsnum *ENDDO

PAGE 89

80Appendix A (Continued) !*DO,step,-24,-30,-1 !theta=1*step !DK,2,ROTY,theta*PI/180 !lsnum=lsnum+1 !LSWRITE,lsnum !*ENDDO !*DO,step,-30,-180,-1 !theta=1*step !DK,2,ROTY,theta*PI/180 !lsnum=lsnum+1 !LSWRITE,lsnum !*ENDDO *Do,nn,1,lsnum LSSOLVE,nn /output,progress,txt,,append *VWRITE,nn Writes data to a file in a formatted sequence %16.8G /output *enddo /STATUS,SOLU FINISH !******************** ********* !******* GET RESULTS ********* !******************** ********* loadSteps=lsnum /POST1 *DIM,rotY2,TABLE,loadSteps *DIM,disX3,TABLE,loadSteps *DIM,disY3,TABLE,loadSteps *DIM,disZ3,TABLE,loadSteps *DIM,disX5,TABLE,loadSteps *DIM,disY5,TABLE,loadSteps *DIM,disZ5,TABLE,loadSteps !*DIM,fx1,TABLE,loadSteps

PAGE 90

81Appendix A (Continued) !*DIM,fy1,TABLE,loadSteps !*DIM,fz1,TABLE,loadSteps !*DIM,mx1,TABLE,loadSteps !*DIM,mz1,TABLE,loadSteps *DIM,momy2,TABLE,loadSteps !*DIM,fx2,TABLE,loadSteps !*DIM,fy2,TABLE,loadSteps !*DIM,fz2,TABLE,loadSteps !*DIM,mx2,TABLE,loadSteps !*DIM,my2,TABLE,loadSteps !*DIM,mz2,TABLE,loadSteps !*DIM,fx3,TABLE,loadSteps !*DIM,fy3,TABLE,loadSteps !*DIM,fz3,TABLE,loadSteps !*DIM,mx3,TABLE,loadSteps !*DIM,my3,TABLE,loadSteps !*DIM,mz3,TABLE,loadSteps !*DIM,fx5,TABLE,loadSteps !*DIM,fy5,TABLE,loadSteps !*DIM,fz5,TABLE,loadSteps !*DIM,mx5,TABLE,loadSteps !*DIM,my5,TABLE,loadSteps !*DIM,mz5,TABLE,loadSteps *Do,nn,1,lsnum set,nn *GET,roty,Node,nkp2,ROT,Y *SET,rotY2(nn),roty *GET,my2,Node,nkp2,RF,MY *SET,MOMY2(nn),my2 *ENDDO /output,output_arc%arclengt h%_asp%aspect%,txt,, *MSG,INFO,'t','w','R','E','arclength' Writes an output message via the ANSYS message subroutine %-8C %-8C %-8C %-8C %-8C

PAGE 91

82Appendix A (Continued) *VWRITE,h2,b2,R,E2,arclength Writes data to a file in a formatted sequence %16.8G %-16.8G %-16 .8G %-16.8G %-16.8G *MSG,INFO,'roty2','my1','my2' %-8C %-8C *VWRITE,rotY2(1),MY1(1),MOMY2(1) %16.8G %-16.8G %-16.8G /output FINISH !*ENDDO !*ENDDO

PAGE 92

83Appendix B: Output ANSYS Text Files The following files are text files outputt ed by ANSYS when running the previous batch files Output to batch 1: t w R E arclengt 2.0000000 5.0000000 313.38000 roty2 my1 -1.74532925E-02 0.0000000 -223.33658 -3.49065850E-02 0.0000000 -446.66746 -5.23598776E-02 0.0000000 -669.98624 -6.98131701E-02 0.0000000 -893.28688 -8.72664626E-02 0.0000000 -1116.5633 -0.10471976 0. 0000000 -1339.8094 -0.12217305 0. 0000000 -1563.0190 -0.13962634 0. 0000000 -1786.1859 -0.15707963 0. 0000000 -2009.3040 -0.17453293 0. 0000000 -2232.3668 -0.19198622 0. 0000000 -2455.3680 -0.20943951 0. 0000000 -2678.3012 -0.22689280 0. 0000000 -2901.1599 -0.26179939 0. 0000000 -3346.6269

PAGE 93

84Appendix B (continued) -0.27925268 0. 0000000 -3569.2216 -0.29670597 0. 0000000 -3791.7147 -0.31415927 0. 0000000 -4014.0989 -0.33161256 0. 0000000 -4236.3671 -0.34906585 0. 0000000 -4458.5119 -0.36651914 0. 0000000 -4680.5257 -0.38397244 0. 0000000 -4902.4008 -0.40142573 0. 0000000 -5124.1293 -0.41887902 0. 0000000 -5345.7032 -0.43633231 0. 0000000 -5567.1141 -0.45378561 0. 0000000 -5788.3536 -0.47123890 0. 0000000 -6009.4127 -0.48869219 0. 0000000 -6230.2827 -0.50614548 0. 0000000 -6450.9542 -0.52359878 0. 0000000 -6671.4176 -0.54105207 0. 0000000 -6891.6632 -0.55850536 0. 0000000 -7111.6808 -0.57595865 0. 0000000 -7331.4599 -0.59341195 0. 0000000 -7550.9898 -0.61086524 0. 0000000 -7770.2592 -0.62831853 0. 0000000 -7989.2566

PAGE 94

85 Appendix B (continued) -0.64577182 0. 0000000 -8207.9700 -0.66322512 0. 0000000 -8426.3871 -0.68067841 0. 0000000 -8644.4949 -0.69813170 0. 0000000 -8862.2802 -0.71558499 0. 0000000 -9079.7292 -0.73303829 0. 0000000 -9296.8274 -0.75049158 0. 0000000 -9513.5600 -0.76794487 0. 0000000 -9729.9115 -0.78539816 0. 0000000 -9945.8657 -0.80285146 0. 0000000 -10161.406 -0.82030475 0. 0000000 -10376.515 -0.83775804 0. 0000000 -10591.174 -0.85521133 0. 0000000 -10805.364 -0.87266463 0. 0000000 -11019.066 -0.89011792 0. 0000000 -11232.258 -0.90757121 0. 0000000 -11444.921 -0.92502450 0. 0000000 -11657.029 -0.94247780 0. 0000000 -11868.562 -0.95993109 0. 0000000 -12079.493 -0.97738438 0. 0000000 -12289.797 -0.99483767 0. 0000000 -12499.447

PAGE 95

86Appendix B (continued) -1.0122910 0. 0000000 -12708.415 -1.0297443 0. 0000000 -12916.672 -1.0471975 0. 0000000 -13124.186 -1.0646508 0. 0000000 -13330.926 -1.0821041 0. 0000000 -13536.856 -1.0995574 0. 0000000 -13741.943 -1.1170107 0. 0000000 -13946.148 -1.1344640 0. 0000000 -14149.431 -1.1519173 0. 0000000 -14351.753 -1.1693706 0. 0000000 -14553.069 -1.1868239 0. 0000000 -14753.333 -1.2042772 0. 0000000 -14952.499 -1.2217305 0. 0000000 -15150.516 -1.2391838 0. 0000000 -15347.329 -1.2566371 0. 0000000 -15542.885 -1.2740903 0. 0000000 -15737.123 -1.2915436 0. 0000000 -15929.982 -1.3089969 0. 0000000 -16121.396 -1.3264502 0. 0000000 -16311.297 -1.3439035 0. 0000000 -16499.612 -1.3613568 0. 0000000 -16686.263

PAGE 96

87Appendix B (continued) -1.3788101 0. 0000000 -16871.171 -1.3962634 0. 0000000 -17054.248 -1.4137167 0. 0000000 -17235.406 -1.4311700 0. 0000000 -17414.548 -1.4486233 0. 0000000 -17591.573 -1.4660766 0. 0000000 -17766.375 -1.4835299 0. 0000000 -17938.840 -1.5009832 0. 0000000 -18108.849 -1.5184364 0. 0000000 -18276.275 -1.5358897 0. 0000000 -18440.984 -1.5533430 0. 0000000 -18602.834 -1.5707963 0. 0000000 -18761.676 -1.5882496 0. 0000000 -18917.349 -1.6057029 0. 0000000 -19069.685 -1.6231562 0. 0000000 -19218.505 -1.6406095 0. 0000000 -19363.619 -1.6580628 0. 0000000 -19504.827 -1.6755161 0. 0000000 -19641.916 -1.6929694 0. 0000000 -19774.659 -1.7104227 0. 0000000 -19902.817 -1.7278760 0. 0000000 -20026.136

PAGE 97

88Appendix B (continued) -1.7453292 0. 0000000 -20144.347 -1.7627825 0. 0000000 -20257.165 -1.7802358 0. 0000000 -20364.287 -1.7976891 0. 0000000 -20465.393 -1.8151424 0. 0000000 -20560.143 -1.8325957 0. 0000000 -20648.179 -1.8500490 0. 0000000 -20729.119 -1.8675023 0. 0000000 -20802.562 -1.8849556 0. 0000000 -20868.081 -1.9024089 0. 0000000 -20925.226 -1.9198622 0. 0000000 -20973.523 -1.9373155 0. 0000000 -21012.470 -1.9547688 0. 0000000 -21041.538 -1.9722220 0. 0000000 -21060.171 -1.9896753 0. 0000000 -21067.780 -2.0071286 0. 0000000 -21063.752 -2.0245819 0. 0000000 -21047.437 -2.0420352 0. 0000000 -21013.354 -2.0594885 0. 0000000 -20969.903 -2.0769418 0. 0000000 -20911.976 -2.0943951 0. 0000000 -20838.789

PAGE 98

89Appendix B (continued) -2.1116863 20750.000 -20838.789 -2.1491990 20500.000 -20838.789 -2.1779545 20250.000 -20838.789 -2.2018684 20000.000 -20838.789 -2.2225688 19750.000 -20838.789 -2.2409992 19500.000 -20838.789 -2.2576922 19250.000 -20838.789 -2.2730046 19000.000 -20838.789 -2.2871865 18750.000 -20838.789 -2.3004208 18500.000 -20838.789 -2.3128456 18250.000 -20838.789 -2.3245683 18000.000 -20838.789 -2.3356739 17750.000 -20838.789 -2.3462310 17500.000 -20838.789 -2.3562959 17250.000 -20838.789 -2.3659153 17000.000 -20838.789 -2.3751281 16750.000 -20838.789 -2.3839673 16500.000 -20838.789 -2.3926752 16250.000 -20838.789 -2.4008369 16000.000 -20838.789 -2.4086915 15750.000 -20838.789

PAGE 99

90Appendix B (continued) -2.4162653 15500.000 -20838.789 -2.4235734 15250.000 -20838.789 -2.4306291 15000.000 -20838.789 -2.4374441 14750.000 -20838.789 -2.4440285 14500.000 -20838.789 -2.4503913 14250.000 -20838.789 -2.4565402 14000.000 -20838.789 -2.4624818 13750.000 -20838.789 -2.4682222 13500.000 -20838.789 -2.4737662 13250.000 -20838.789 -2.4791184 13000.000 -20838.789 -2.4842823 12750.000 -20838.789 -2.4892613 12500.000 -20838.789 -2.4940578 12250.000 -20838.789 -2.4986743 12000.000 -20838.789 -2.5031124 11750.000 -20838.789 -2.5073737 11500.000 -20838.789 -2.5114593 11250.000 -20838.789 -2.5153699 11000.000 -20838.789 -2.5191064 10750.000 -20838.789 -2.5226689 10500.000 -20838.789

PAGE 100

91Appendix B (continued) -2.5260578 10250.000 -20838.789 -2.5292730 10000.000 -20838.789 -2.5323146 9750.0000 -20838.789 -2.5351823 9500.0000 -20838.789 -2.5378759 9250.0000 -20838.789 -2.5403950 9000.0000 -20838.789 -2.5427394 8750.0000 -20838.789 -2.5449086 8500.0000 -20838.789 -2.5469023 8250.0000 -20838.789 -2.5487202 8000.0000 -20838.789 -2.5503619 7750.0000 -20838.789 -2.5518274 7500.0000 -20838.789 -2.5531163 7250.0000 -20838.789 -2.5542287 7000.0000 -20838.789 -2.5551646 6750.0000 -20838.789 -2.5559241 6500.0000 -20838.789 -2.5565074 6250.0000 -20838.789 -2.5569150 6000.0000 -20838.789 -2.5571473 5750.0000 -20838.789 -2.5572050 5500.0000 -20838.789 -2.5570888 5250.0000 -20838.789

PAGE 101

92Appendix B (continued) -2.5567996 5000.0000 -20838.789 -2.5563384 4750.0000 -20838.789 -2.5557065 4500.0000 -20838.789 -2.5549050 4250.0000 -20838.789 -2.5539353 4000.0000 -20838.789 -2.5527990 3750.0000 -20838.789 -2.5514978 3500.0000 -20838.789 -2.5500332 3250.0000 -20838.789 -2.5484073 3000.0000 -20838.789 -2.5466218 2750.0000 -20838.789 -2.5446788 2500.0000 -20838.789 -2.5425803 2250.0000 -20838.789 -2.5403286 2000.0000 -20838.789 -2.5379256 1750.0000 -20838.789 -2.5353738 1500.0000 -20838.789 -2.5326754 1250.0000 -20838.789 -2.5298326 1000.0000 -20838.789 -2.5268478 750.00000 -20838.789 -2.5237233 500.00000 -20838.789 -2.5204615 250.00000 -20838.789 -2.5170647 0. 0000000 -20838.789

PAGE 102

93Appendix B (continued) -2.5135352 250. 00000 -20838.789 -2.5098753 500.00000 -20838.789 -2.5060873 750.00000 -20838.789 -2.5021736 1000.0000 -20838.789 -2.4981362 1250.0000 -20838.789 -2.4939775 1500.0000 -20838.789 -2.4896995 1750.0000 -20838.789 -2.4853045 2000.0000 -20838.789 -2.4807944 2250.0000 -20838.789 -2.4761714 2500.0000 -20838.789 -2.4714372 2750.0000 -20838.789 -2.4665939 3000.0000 -20838.789 -2.4616434 3250.0000 -20838.789 -2.4565874 3500.0000 -20838.789 -2.4514276 3750.0000 -20838.789 -2.4461657 4000.0000 -20838.789 -2.4408034 4250.0000 -20838.789 -2.4353422 4500.0000 -20838.789 -2.4297836 4750.0000 -20838.789 -2.4241290 5000.0000 -20838.789 -2.4183798 5250.0000 -20838.789

PAGE 103

94Appendix B (continued) -2.4125372 5500. 0000 -20838.789 -2.4066026 5750.0000 -20838.789 -2.4005771 6000.0000 -20838.789 -2.3944618 6250.0000 -20838.789 -2.3882578 6500.0000 -20838.789 -2.3819660 6750.0000 -20838.789 -2.3755875 7000.0000 -20838.789 -2.3691230 7250.0000 -20838.789 -2.3625734 7500.0000 -20838.789 -2.3559395 7750.0000 -20838.789 -2.3492219 8000.0000 -20838.789 -2.3424212 8250.0000 -20838.789 -2.3355382 8500.0000 -20838.789 -2.3285733 8750.0000 -20838.789 -2.3215270 9000.0000 -20838.789 -2.3143997 9250.0000 -20838.789 -2.3071917 9500.0000 -20838.789 -2.2999036 9750.0000 -20838.789 -2.2925354 10000.000 -20838.789 -2.2850874 10250.000 -20838.789 -2.2775598 10500.000 -20838.789

PAGE 104

95Appendix B (continued) -2.2699527 10750. 000 -20838.789 -2.2622662 11000.000 -20838.789 -2.2545004 11250.000 -20838.789 -2.2466552 11500.000 -20838.789 -2.2387305 11750.000 -20838.789 -2.2307263 12000.000 -20838.789 -2.2226425 12250.000 -20838.789 -2.2144788 12500.000 -20838.789 -2.2062350 12750.000 -20838.789 -2.1979108 13000.000 -20838.789 -2.1895059 13250.000 -20838.789 -2.1810201 13500.000 -20838.789 -2.1724528 13750.000 -20838.789 -2.1638037 14000.000 -20838.789 -2.1550723 14250.000 -20838.789 -2.1462580 14500.000 -20838.789 -2.1373605 14750.000 -20838.789 -2.1283789 15000.000 -20838.789 -2.1193129 15250.000 -20838.789 -2.1101616 15500.000 -20838.789 -2.1009245 15750.000 -20838.789

PAGE 105

96Appendix B (continued) -2.0916007 16000. 000 -20838.789 -2.0821896 16250.000 -20838.789 -2.0726903 16500.000 -20838.789 -2.0631020 16750.000 -20838.789 -2.0534238 17000.000 -20838.789 -2.0436550 17250.000 -20838.789 -2.0337944 17500.000 -20838.789 -2.0238412 17750.000 -20838.789 -2.0137943 18000.000 -20838.789 -2.0036528 18250.000 -20838.789 -1.9934155 18500.000 -20838.789 -1.9830814 18750.000 -20838.789 -1.9726493 19000.000 -20838.789 -1.9621180 19250.000 -20838.789 -1.9514863 19500.000 -20838.789 -1.9407530 19750.000 -20838.789 -1.9299167 20000.000 -20838.789 -1.9189761 20250.000 -20838.789 -1.9079298 20500.000 -20838.789 -1.8967765 20750.000 -20838.789 -1.8855145 21000.000 -20838.789

PAGE 106

97Appendix B (continued) -1.8741424 21250. 000 -20838.789 -1.8626585 21500.000 -20838.789 -1.8510613 21750.000 -20838.789 -1.8393489 22000.000 -20838.789 -1.8275197 22250.000 -20838.789 -1.8155716 22500.000 -20838.789 -1.8035027 22750.000 -20838.789 -1.7913110 23000.000 -20838.789 -1.7789943 23250.000 -20838.789 -1.7665503 23500.000 -20838.789 -1.7539766 23750.000 -20838.789 -1.7414670 24000.000 -20838.789 -1.7286321 24250.000 -20838.789 -1.7153217 24500.000 -20838.789 -1.7023682 24750.000 -20838.789 -1.6892914 25000.000 -20838.789 -1.6758923 25250.000 -20838.789 -1.6623305 25500.000 -20838.789 -1.6486299 25750.000 -20838.789 -1.6347536 26000.000 -20838.789 -1.6207163 26250.000 -20838.789

PAGE 107

98Appendix B (continued) -1.6065086 26500.000 -20838.789 -1.5921240 26750.000 -20838.789 -1.5775557 27000.000 -20838.789 -1.5627838 27250.000 -20838.789 -1.5478262 27500.000 -20838.789 -1.5326576 27750.000 -20838.789 -1.5172685 28000.000 -20838.789 -1.5016464 28250.000 -20838.789 -1.4857772 28500.000 -20838.789 -1.4696448 28750.000 -20838.789 -1.4532304 29000.000 -20838.789 -1.4365121 29250.000 -20838.789 -1.4194642 29500.000 -20838.789 -1.4020562 29750.000 -20838.789 -1.3842513 30000.000 -20838.789 -1.3660050 30250.000 -20838.789 -1.3472626 30500.000 -20838.789 -1.3279554 30750.000 -20838.789 -1.3079957 31000.000 -20838.789 -1.2872692 31250.000 -20838.789 -1.2656218 31500.000 -20838.789

PAGE 108

99Appendix B (continued) -1.2428387 31750. 000 -20838.789 -1.2186097 32000.000 -20838.789 -1.1924378 32250.000 -20838.789 -1.1634605 32500.000 -20838.789 -1.1300314 32750.000 -20838.789 -1.0876665 33000.000 -20838.789 -0.96938944 33250. 000 -20838.789 -0.97738438 0. 0000000 33239.297 -0.95993109 0. 0000000 33216.962 -0.94247780 0. 0000000 33183.262 -0.92502450 0. 0000000 33137.183 -0.90757121 0. 0000000 33080.835 -0.89011792 0. 0000000 33016.772 -0.87266463 0. 0000000 32947.893 -0.85521133 0. 0000000 32877.319 -0.83775804 0. 0000000 32808.231 -0.82030475 0. 0000000 32743.724 -0.80285146 0. 0000000 32686.672 -0.78539816 0. 0000000 32639.630 -0.76794487 0. 0000000 32604.778 -0.75049158 0. 0000000 32583.894

PAGE 109

100Appendix B (continued) -0.69813170 0. 0000000 32617.286 -0.68067841 0. 0000000 32662.951 -0.66322512 0. 0000000 32726.546 -0.64577182 0. 0000000 32808.225 -0.62831853 0. 0000000 32908.048 -0.61086524 0. 0000000 33026.023 -0.59341195 0. 0000000 33162.150 -0.57595865 0. 0000000 33316.455 -0.55850536 0. 0000000 33489.036 -0.54105207 0. 0000000 33680.107 -0.52359878 0. 0000000 33890.058 -0.50614548 0. 0000000 34119.538 -0.48869219 0. 0000000 34369.588 -0.47123890 0. 0000000 34641.856 -0.45378561 0. 0000000 34939.009 -0.43633231 0. 0000000 35265.577 -0.41887902 0. 0000000 35630.013 -0.40142573 0. 0000000 36049.949 -0.38397244 0. 0000000 36595.985 -0.38554372 36500. 000 36595.985 -0.38585273 36525. 000 36595.985

PAGE 110

101Appendix B (continued) -0.38515289 36550. 000 36595.985 -0.38449126 36575. 000 36595.985 -0.38384641 36600. 000 36595.985 -0.38321915 36625. 000 36595.985 -0.38260998 36650. 000 36595.985 -0.38201942 36675. 000 36595.985 -0.38144804 36700. 000 36595.985 -0.38089643 36725. 000 36595.985 -0.38036523 36750. 000 36595.985 -0.37985513 36775. 000 36595.985 -0.37936686 36800. 000 36595.985 -0.37890120 36825. 000 36595.985 -0.37845901 36850. 000 36595.985 -0.37804118 36875. 000 36595.985 -0.37764871 36900. 000 36595.985 -0.37728264 36925. 000 36595.985 -0.37694414 36950. 000 36595.985 -0.37663445 36975. 000 36595.985 -0.37635492 37000. 000 36595.985 -0.37610705 37025. 000 36595.985 -0.37589246 37050. 000 36595.985

PAGE 111

102Appendix B (continued) -0.37571295 37075. 000 36595.985 -0.37557050 37100. 000 36595.985 -0.37546730 37125. 000 36595.985 -0.37540582 37150. 000 36595.985 -0.37538881 37175. 000 36595.985 -0.37541939 37200. 000 36595.985 -0.37550113 37225. 000 36595.985 -0.37563814 37250. 000 36595.985 -0.37583525 37275. 000 36595.985 -0.37609820 37300. 000 36595.985 -0.37643398 37325. 000 36595.985 -0.37685140 37350. 000 36595.985 -0.37736199 37375. 000 36595.985 -0.37798165 37400. 000 36595.985 -0.37873359 37425. 000 36595.985 -0.37965267 37450. 000 36595.985 -0.38010927 37475. 000 36595.985 -0.38116689 37500. 000 36595.985 -0.38238823 37525. 000 36595.985 -0.38314500 37550. 000 36595.985 -0.38489833 37575. 000 36595.985

PAGE 112

103Appendix B (continued) -0.38681297 37600. 000 36595.985 -0.38913579 37625. 000 36595.985 -0.39195722 37650. 000 36595.985 -0.39550168 37675. 000 36595.985 -0.40020412 37700. 000 36595.985 -0.40714338 37725. 000 36595.985 -0.40666172 0. 0000000 37712.666 -0.40840705 0. 0000000 37716.816 -0.41015237 0. 0000000 37719.958 -0.41189770 0. 0000000 37722.498 -0.41364303 0. 0000000 37724.462 -0.41538836 0. 0000000 37725.891 -0.41713369 0. 0000000 37726.819 -0.41887902 0. 0000000 37727.276 -0.41887902 0. 0000000 37727.081 -0.43633231 0. 0000000 37710.977 -0.45378561 0. 0000000 37667.193 -0.47123890 0. 0000000 37604.736 -0.48869219 0. 0000000 37528.474 -0.50614548 0. 0000000 37441.458 -0.52359878 0. 0000000 37345.734

PAGE 113

104Appendix B (continued) -0.52359878 0. 0000000 37345.734 -0.54105207 0. 0000000 37242.754 -0.55850536 0. 0000000 37133.583 -0.57595865 0. 0000000 37019.033 -0.59341195 0. 0000000 36899.733 -0.61086524 0. 0000000 36776.183 -0.62831853 0. 0000000 36648.787 -0.64577182 0. 0000000 36517.876 -0.66322512 0. 0000000 36383.722 -0.68067841 0. 0000000 36246.555 -0.69813170 0. 0000000 36106.693 -0.71558499 0. 0000000 35964.045 -0.73303829 0. 0000000 35818.884 -0.75049158 0. 0000000 35671.327 -0.76794487 0. 0000000 35521.477 -0.78539816 0. 0000000 35369.425 -0.80285146 0. 0000000 35215.250 -0.82030475 0. 0000000 35061.914 -0.83775804 0. 0000000 34903.497 -0.85521133 0. 0000000 34743.280 -0.87266463 0. 0000000 34581.162

PAGE 114

105Appendix B (continued) -0.89011792 0. 0000000 34417.187 -0.90757121 0. 0000000 34251.387 -0.92502450 0. 0000000 34083.793 -0.94247780 0. 0000000 33914.428 -0.95993109 0. 0000000 33743.315 -0.97738438 0. 0000000 33570.470 -0.99483767 0. 0000000 33395.908 -1.0122910 0. 0000000 33219.641 -1.0297443 0. 0000000 33041.678 -1.0471975 0. 0000000 32862.027 -1.0646508 0. 0000000 32680.692 -1.0821041 0. 0000000 32497.675 -1.0995574 0. 0000000 32312.978 -1.1170107 0. 0000000 32126.600 -1.1344640 0. 0000000 31938.538 -1.1519173 0. 0000000 31748.789 -1.1693706 0. 0000000 31557.348 -1.1868239 0. 0000000 31364.208 -1.2042772 0. 0000000 31169.364 -1.2217305 0. 0000000 30972.805 -1.2391838 0. 0000000 30774.524

PAGE 115

106Appendix B (continued) -1.2566371 0. 0000000 30574.511 -1.2740903 0. 0000000 30372.755 -1.2915436 0. 0000000 30169.246 -1.3089969 0. 0000000 29963.972 -1.3264502 0. 0000000 29756.922 -1.3439035 0. 0000000 29548.085 -1.3613568 0. 0000000 29337.447 -1.3788101 0. 0000000 29124.999 -1.3962634 0. 0000000 28910.728 -1.4137167 0. 0000000 28694.622 -1.4311700 0. 0000000 28476.671 -1.4486233 0. 0000000 28256.864 -1.4660766 0. 0000000 28035.190 -1.4835299 0. 0000000 27811.641 -1.5009832 0. 0000000 27586.207 -1.5184364 0. 0000000 27358.880 -1.5358897 0. 0000000 27129.653 -1.5533430 0. 0000000 26898.521 -1.5707963 0. 0000000 26665.477 -1.5882496 0. 0000000 26430.518 -1.6057029 0. 0000000 26193.642

PAGE 116

107Appendix B (continued) -1.6231562 0. 0000000 25954.846 -1.6406095 0. 0000000 25714.131 -1.6580628 0. 0000000 25471.497 -1.6755161 0. 0000000 25226.948 -1.6929694 0. 0000000 24980.487 -1.7104227 0. 0000000 24732.120 -1.7278760 0. 0000000 24481.854 -1.7453292 0. 0000000 24229.697 -1.7627825 0. 0000000 23975.659 -1.7802358 0. 0000000 23719.751 -1.7976891 0. 0000000 23461.986 -1.8151424 0. 0000000 23202.379 -1.8325957 0. 0000000 22940.945 -1.8500490 0. 0000000 22677.701 -1.8675023 0. 0000000 22412.665 -1.8849556 0. 0000000 22145.856 -1.9024089 0. 0000000 21877.296 -1.9198622 0. 0000000 21607.006 -1.9373155 0. 0000000 21335.009 -1.9547688 0. 0000000 21061.329 -1.9722220 0. 0000000 20785.989

PAGE 117

108 Appendix B (continued) -2.0071286 0. 0000000 20230.438 -2.0245819 0. 0000000 19950.279 -2.0420352 0. 0000000 19668.568 -2.0594885 0. 0000000 19385.333 -2.0769418 0. 0000000 19100.602 -2.0943951 0. 0000000 18814.405 -2.1118484 0. 0000000 18526.770 -2.1293017 0. 0000000 18237.727 -2.1467550 0. 0000000 17947.306 -2.1642083 0. 0000000 17655.537 -2.1816616 0. 0000000 17362.450 -2.1991149 0. 0000000 17068.074 -2.2165681 0. 0000000 16772.440 -2.2340214 0. 0000000 16475.578 -2.2514747 0. 0000000 16177.516 -2.2689280 0. 0000000 15878.286 -2.2863813 0. 0000000 15577.915 -2.3038346 0. 0000000 15276.435 -2.3212879 0. 0000000 14973.873 -2.3387412 0. 0000000 14670.258 -2.3561945 0. 0000000 14365.620

PAGE 118

109Appendix B (continued) -2.3911011 0. 0000000 13753.382 -2.4085544 0. 0000000 13445.840 -2.4260077 0. 0000000 13137.384 -2.4434609 0. 0000000 12828.041 -2.4609143 0. 0000000 12517.840 -2.4783675 0. 0000000 12206.805 -2.4958208 0. 0000000 11894.963 -2.5132741 0. 0000000 11582.339 -2.5307274 0. 0000000 11268.959 -2.5481807 0. 0000000 10954.847 -2.5656340 0. 0000000 10640.029 -2.5830873 0. 0000000 10324.528 -2.6005406 0. 0000000 10008.369 -2.6179939 0. 0000000 9691.5748 -2.6354472 0. 0000000 9374.1695 -2.6529005 0. 0000000 9056.1761 -2.6703538 0. 0000000 8737.6170 -2.6878071 0. 0000000 8418.5154 -2.7052603 0. 0000000 8098.8932 -2.7227136 0. 0000000 7778.7726 -2.7401669 0. 0000000 7458.1751

PAGE 119

110Appendix B (continued) -2.7576202 0. 0000000 7137.1227 -2.7750735 0. 0000000 6815.6366 -2.7925268 0. 0000000 6493.7379 -2.8099801 0. 0000000 6171.4474 -2.8274334 0. 0000000 5848.7862 -2.8448867 0. 0000000 5525.7746 -2.8623400 0. 0000000 5202.4332 -2.8797933 0. 0000000 4878.7818 -2.8972466 0. 0000000 4554.8410 -2.9146998 0. 0000000 4230.6305 -2.9321531 0. 0000000 3906.1702 -2.9496064 0. 0000000 3581.4796 -2.9670597 0. 0000000 3256.5785 -2.9845130 0. 0000000 2931.4863 -3.0019663 0. 0000000 2606.2224 -3.0194196 0. 0000000 2280.8059 -3.0368729 0. 0000000 1955.2564 -3.0543262 0. 0000000 1629.5929 -3.0717795 0. 0000000 1303.8345 -3.0892328 0. 0000000 978.00002 -3.1066861 0. 0000000 652.10892

PAGE 120

111Appendix B (continued) -3.1241394 0. 0000000 326.18001 -3.1415927 0.0000000 0.23225880

PAGE 121

112Appendix C: MatLab M-File The following file is an M-f ile that is used in MatLab. It reads the output of the ANSYS Batch files. M-File: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Ansys data analysis file % % For an Ansys batch file % % which produces an output file named knee_output.txt % % Version 1: May 18,2007 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% filename = [ 'output_arc%arclength%_asp%aspect%.txt' ]; string1 = [ 'C:\DOCUME~1\JOSEPH~1\DESKTOP\MYMAST~1\ANSYS\A7_Optimized\Run5_7square s_optimized\' ]; fid1 = fopen([string1,filename]); % opens the file ABT = fread(fid1); % reads the file into variable ABT fclose(fid1); %closes the data file GBT = native2unicode(ABT)'; %changes data from machine code to text s_iB = findstr( 'my1' GBT); % finds end of header A=str2num(GBT(s_iB+4:end)) % turns the data into a numerical matrix roty2 = A(:,1); my2 = A(:,2); my3 = A(:,3) figure(1) plot(roty2*180/pi,[my2,my3], '*' )


xml version 1.0 encoding UTF-8 standalone no
record xmlns http:www.loc.govMARC21slim xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.loc.govstandardsmarcxmlschemaMARC21slim.xsd
leader nam Ka
controlfield tag 001 001935151
003 fts
005 20080421154801.0
006 m||||e|||d||||||||
007 cr mnu|||uuuuu
008 080421s2007 flua sbm 000 0 eng d
datafield ind1 8 ind2 024
subfield code a E14-SFE0002290
040
FHM
c FHM
035
(OCoLC)225866603
049
FHMM
090
TJ145 (ONLINE)
1 100
Choueifati, Joseph Georges.
0 245
Design and modeling of a bistable spherical compliant micromechanism
h [electronic resource] /
by Joseph Georges Choueifati.
260
[Tampa, Fla.] :
b University of South Florida,
2007.
3 520
ABSTRACT: Compliant bistable mechanisms are mechanisms that have two stable equilibrium positions within their range of motion. Their bistability is mainly due to the elasticity of their members. This thesis introduces a new type of bistable micromechanisms, the Bistable, Spherical, Compliant, four-bar Micromechanism (BSCM). Theory to predict bistable positions and configurations is also developed. Bistabilty was demonstrated through testing done on micro-prototypes. Compared to the mathematical model of the BSCM, Finite element models of the BSCM indicated important qualitative differences in the mechanism's stability behavior and its input-angle-input torque relation. The BSCM has many valuable features, such as: Two stable positions that require power only when moving from one stable position to the other, accurate and repeatable out-of-plane motion with resistance to small perturbations. The BSCM may be useful in several applications such as active Braille systems and Digital Light Processing (DLP) chips.
502
Thesis (M.S.)--University of South Florida, 2007.
504
Includes bibliographical references.
516
Text (Electronic thesis) in PDF format.
538
System requirements: World Wide Web browser and PDF reader.
Mode of access: World Wide Web.
500
Title from PDF of title page.
Document formatted into pages; contains 112 pages.
590
Advisor: Craig Lusk, Ph.D.
653
Bistable.
Compliant.
Spherical.
690
Dissertations, Academic
z USF
x Mechanical Engineering
Masters.
773
t USF Electronic Theses and Dissertations.
4 856
u http://digital.lib.usf.edu/?e14.2290