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Choueifati, Joseph Georges.
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Design and modeling of a bistable spherical compliant micromechanism
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by Joseph Georges Choueifati.
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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, fourbar Micromechanism (BSCM). Theory to predict bistable positions and configurations is also developed. Bistabilty was demonstrated through testing done on microprototypes. 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 inputangleinput 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 outofplane 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.
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Thesis (M.S.)University of South Florida, 2007.
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Includes bibliographical references.
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Text (Electronic thesis) in PDF format.
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System requirements: World Wide Web browser and PDF reader.
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Advisor: Craig Lusk, Ph.D.
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Bistable.
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Dissertations, Academic
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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
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Dedication To my Father, Georges, my Mother, Amal, my Brother, Jules, and my Sister, Joelle.
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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.
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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 OrthoPlanar 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 FourBar Mechanism 9 3.1.1.1 ClosedForm Equations 10 3.1.2 PseudoRigidBody Model 11 3.1.2.1 Small Length Flexural Pivot 12 3.1.2.2 PRBM FourBar Mechanism 14 3.1.3 Definition of Bistability 16 3.2 Spherical Mechanisms 17 3.2.1 Spherical Trigonometry 18 3.2.2 Spherical FourBar Mechanism 21 Chapter 4: Bistability of a Spherical FourBar Mechanism 24 4.1 Principle of Virtual Work 24 4.2 Virtual Work Equation of Compliant Spherical FourBar Mechanism 25
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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 FiniteElement 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 MFile 112
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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
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iv List of Figures Figure 2.1 OrthoPlanar Spherical Mechanism 5 Figure 2.2 Spherical Bistable Mechanism 5 Figure 2.3 Compliant MEMS 8 Figure 3.1 Rigid Link FourBar CrankRocker Mechanism 10 Figure 3.2 SmallLength Flexural Pivot 12 Figure 3.3 PseudoRigidBody Model 13 Figure 3.4 A Compliant FourBar Mechanism and its PseudoRigid body model 15 Figure 3.5 BallOnTheHill 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 Â“CÂ” 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 FourB 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 FourBar Mechanism in Equilibrium from Joint 1 31 Figure 4.4 Input Moment Required to Hold the Spherical FourBar Mechanism in Equilibrium from Joint 2 33
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v Figure 4.5 Input Moment Required to Hold the Spherical FourBar Mechanism in Equilibrium from Joint 3 35 Figure 4.6 Input Moment Required to Hold the Spherical FourBar 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 OutOfPlane Position 44 Figure 5.6 SEM Images of BSCM in its Second St able Equilibrium Position 45 Figure 5.7 Mathematical Model of the MomentRotation Relationship for a BSCM with Rigid Links and ShortLength Flexural Pivots 46 Figure 5.8 MomentRotation Relationshi p from a FEA of the BSCM Prototype 46
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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 croprototypes. Compared to the mathematical model of t he BSCM, Finite element m odels of the BSCM indicated important qualitative differ ences in the mechanismÂ’s stability behavior and its inputangleinput 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 outofpl 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.
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1 Chapter 1 1.1 Objective The objective of this research wa s to develop a bistable Microelectromechanical 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 microscaled devices that were designed, fabricated and analyzed as part of this research. Finiteelement 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
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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], microoptic al systems [5], it may be us eful for MEMS to achieve accurate threedimensional motion. 1.3 Contribution The research in this thesis provides a new design in the MEMS field. A bistable spherical compliant fourbar 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 mechanismÂ’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 todayÂ’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 fourbar 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.
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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 hingemounted digital micromirror devices (DMD); each of t hese micromirrors measures less than onefifth 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
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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 micromanipulator may oper ate on cells [1]. Briefly, a typical MEMS device can be defined as: (1) device consisting of micromechanisms 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 threedimensional motion. To achieve that goal, researchers are requir ed to come up with creative designs. 2.2.2 OrthoPlanar Mechanisms Orthoplanar (OP) mechanisms [9] ar e a type of mechanism that can achieve outofplane motion. Ort hoPlanar mechanisms are defined as
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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: OrthoPlanar Spherical Mechanism [3] Figure 2.2: Spherical Bistable Mechanism [3]
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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 mechanismÂ’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 outof 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 braillebased 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 micromirrors. According to Fukushige: If a long stroke in the outofplane direction, a large output force, and high integration can be simu ltaneously realized, microoptical systems such as actuation of micromirrors become possible" [5]. This thesis offers a detailed analysis of a MEMS device with large displacement and precise outofplane motion.
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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 multipieces joints with compliant single member joint will likely increase their lifespan [12,13, 14]. On the other hand; the advantages offered by compliant micromechanisms donÂ’t come without challenges. Their dynamic and kinemati cs analysis is difficult but can be simplified using easier techniques such as the Pseudorigidbody model (PRBM) [14]. PRBMs model compliant mechanisms as rigidbody 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.
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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 fourbar MEMS with precise and large out of plane displacement.
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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 fourbar 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 fourbar mechanism Usually, when analyzing a rigidbody 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 fourbar mechanism is presented [21]. 3.1.1 Position Analysis of Planar FourBar Mechanism Many methods have been developed fo r the position and displacement analysis of a fourbar mechanism. In this paper, we will focus on the analytical
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10 method, specifically a closedform so lution for a fourbar crankrocker mechanism. Figure 3.1: Rigid Link FourBar CrankRocker Mechanism [15] 3.1.1.1 ClosedForm Equations Consider the fourbar crankrocker 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)
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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 PseudoRigidBody Model Numerous methods have been develope d to analyze large deflections [22]. The pseudorigidbody model (PRBM) provides a simple approach for the analysis of systems that undergo la rge nonlinear 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 rigidbody components with similar fo rcedeflection characteristics [15]. Different types of mechanisms require diffe rent models; in this thesis we the small length flexural pivot (SLFP) and th e fourbar PRBM. Also, the method of virtual work is a fundamental tool for force deflection behavior of a mechanism.
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12Thus, it can be used to determine the forcedeflection behavior and hence the bistability of a mechanism [18]. 3.1.2.1 SmallLength 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: SmallLength Flexural Pivot [15] The beam is composed of two segm ents: A small flexible segment Â“lÂ” called small length flexural pi vot and a large rigid segment Â“LÂ”. Since Â“lÂ” is flexible and Â“LÂ” is rigid, then: L lEI EI (3.8) Where E the materialÂ’s YoungÂ’s modulus and I the second moment of area.
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13Since the flexible section is a lot s horter than the rigid one, the beamÂ’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: PseudoRigidBody Model The x and y coordinates of the beamÂ’s end are approximated as cos 2 2 l L l a (3.9) And sin 2 l L b (3.10)
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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 FourBar Mechanism A fourbar mechanism with compliant joints can be modeled using the PRBM concept. Figure 3.4 shows a comp liant fourbar mechanism with its pseudorigidbody model. The compliant joints in Figure 3.4a) are replaced by torsional springs in Figure 3.4b). The energy equations governing a comp liant fourbar 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
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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 FourBar Mechanism and Part b) its PseudoRigid Body Model
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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 LagrangeDirichlet 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 ballonthehill analogy Figure 3.5. A sm all impulse applied on to t he ball at eit her Â‘AÂ’ or Â‘DÂ’ will make oscillate but then it will settle back into its original position. Positions Â‘AÂ’ and Â‘DÂ’ would then be considered as stable equilibrium positions, locations where
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17the ball has lowest pot ential energy. At position Â‘BÂ’, the ball is considered to be in unstable equilibrium position because unde r a small disturbance the ball is going to go to either Â‘AÂ’ or Â‘DÂ’ positions. Figure 3.5: BallOnTheHill 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 mechanismÂ’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
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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
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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, uppercase roman le tters represent the dihedral angles between the two planes containing inters ecting great circles; lowercase 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.
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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 NapierÂ’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 Â“CÂ” is 90 The other 5 angles can be drawn into a circle which has been divided into 5 arcs seen in Figure 3.7.
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21 Figure 3.8: Schematic of the Parts of Right Spherical Triangle with Right Angle Â“CÂ” 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 = 900A (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 FourBar Mechanism A spherical fourbar 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.
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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 lowercase greek letters. Figure 3.9: Spherical FourBar 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,
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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.
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24Chapter 4 Bistability of a Spherical FourBar 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. Â“The net virtual work of all active forces is zero if and only if an ideal mechanical sy stem is in equilibrium.Â” 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 pseudorigid 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
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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 FourBar Mechanism Consider the spherical mechanism of Fi gure 3.7 but with four smalllengthflexuralpivots replacing the four revolute joints as shown in Figure 4.1a). The position equations are already develop ed, see equations 3. 223.27.The energy equation, similarly to a planar mechani sm, can be easily developed using the pseudorigidbody model of t he compliant spherical fourbar mechanism (CSM) of Figure 4.1b). 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 nonzero while setting the other springsÂ’ constants equal to zero. Thus a procedure is developed
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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 rBar Mechanism and Its b) PRBM
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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.
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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 (4.5a) Equilibrium positions for a particular torsional spring 2 ,0 d d k Minitial i final i i in (4.5b) The first part of Equation 4.5b, 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 4.5b, 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.
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294.3 A Simplified Mathematical Model of a Bistable CSM In the preceding section, a procedure for determining bistability in a general spherical compliant fourbar mechanism was developed. In this section, a specific fourbarmechanism 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.
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30Recalling equations 4.5a and 4.5b with 2 12 23 34 41 An examination of the momentrotation 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 Â“joint 1Â” 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 momentrotation curve is linear. Thus, the elasticity associated with Â“j oint 1Â” 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
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31 Figure 4.3: Input Moment Required to Hold the Spherical FourBar 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
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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)
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33 Figure 4.4: Input Moment Required to Hold the Spherical FourBar Mechanism in Equilibrium for Joint 2 The momentrotation 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
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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 Â–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
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35its first stable equilibrium position. When the input link r2 is rotated by 90o, the mechanism reaches its nonstable 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 FourBar Bechanism in Equilibrium for Joint 3 A B C D E
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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)
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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 FourBar Mechanism in Equilibrium for Joint 4
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38 The momentrotation 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 momentrotati on would then have the similar nonlinear 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 fourbar micromechanism is described based on t he analysis presented in this chapter.
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39Chapter 5 A Bistable Spherical Co mpliant Micromechanism Micromechanisms with accurate outoffplane 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 outofplane. 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 MultiUser MEMS Processes (MUMPs). MUMPs is a threelayer polysilicon surface micromachining process. Figure 5.1 shows a cross section of the threelayer 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
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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
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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 sphericalf 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
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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 outofplane 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
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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
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44 Figure 5.5: SEM Image of the BSCM in an OutOfPlane 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 microprobes 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
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45As mentioned before the BSCM is in its first stable equilibrium position when it is in its Â“as fabricated positionÂ” 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 FiniteElement 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 mechanismÂ’s links and not just at the short length flexural pi vots. The increased flexibility resulted in important qualitative differ ences in the mechanismÂ’s stability behavior and its
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46input angleinput 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 mentRotation Relationship for a BSCM with Rigid Links and ShortLength Flexural Pivots Figure 5.8: MomentRotation Relationship fr om a FEA of the BSCM Prototype
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47 The momentrotation 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.
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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 microprototype. Compared to the PRBM with small length flexural pivots of the BSCM, Finiteelement models of the BSCM indicated im portant qualitative difference in the mechanismÂ’s stabilit y behavior and its inputangleinput moment relation and that may be due to the defle ction of the Â‘rigid Â’ 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.
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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 microprobes. 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.
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50References [1] K. J. Gabriel, Â“Microelectr omechanical Systems (MEMS) Tutorial ,Â” IEEE Test Conference (TC), pp. 432441, 1998. [2] M. Mehregany and M. Huff, Â“Microelectromec hanical Systems," Proceedings of the IEEE Cornell C onference on Advanced Concepts in High Speed Semiconductor Device s and Circuits, pp. 918, 1995. [3] C. Lusk, Â“OrthoPlanar Mechanisms for Microelctromechanical Systems,Â” 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, Â“MEMS commerci alization: Ingredi ents for success," Proceedings of the IEEE Micro Electr o Mechanical Systems (MEMS), pp. 710, 2000. [7] Texas Instruments, Â“Digit al Light Processors,Â” http://www.dlp.com/tech/what.aspx 2007. [8] J. A. Bradley, Â“Design of Surf ace Micromachined Compliant MEMS,Â” Thesis, Iowa State University, Ames, IA. [9] J. Parise, L. Howell, and S. Magleby, Â“Orthoplanar mechanisms," Proceedings of the 2000 ASME De sign Engineering Technical Conferences, DETC2000/MECH 14193, pp. 115, 2000. [10] Ananthasuresh, GK and Howell, Larry L (2005) Â“Mechanical Design of Compliant MicrosystemsA Pe rspective and ProspectsÂ”. Journal of Mechanical Design 127(4):pp. 736738. [11] Felton, B. Â“Better robots th rough clean livingÂ”. Intec, May 2001.
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51[12] R. Cragun and L. L. Howell, Â“A New Cons trained Thermal Expansion MicroActuator," American Society of Mechani cal Engineers, Dynamic Systems and Control Division (Publication) DSC vol. 66, pp. 365371, 1998. [13] C. D. Lott, J. Harb T. W. McClain, and L. Howe ll, Â“Dynamic Modelling of a SurfaceMicromachined 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. 374377, 2001. [14] Howell, L.L. and Midha, A., 1994, "A Method for the Design of Compliant Mechanisms with SmallLength Flexural Pivots," ASME Journal of Mechanical Design, Vol. 116, No. 1, pp. 280290. [15] L. Howell, Compliant Mechanisms. New York: WileyInterscience, 2001 [16] Howell, L.L., Rao, S.S., and Mi dha, A., 1994, "The ReliabilityBased Optimal Design of a Bistable Compliant Mechanism," ASME Journal of Mechanical Design, Vol. 116, No.4, pp. 11151121. [17] Jensen, B.D., Howell, L.L., Gunyan, D.B., and Salmon, L.G., 1997, "The Design and Analysis of Compliant MEMS Using the PseudoRigidBody Model," Microelectomechanical Systems (MEMS) 1997, presented at the 1997 ASME International Mechanical Engineering Congress and Exposition, November 1621, 1997, Dallas, Texas, DSCVol. 62, pp. 119126. [18] Jensen, B.D., Howell, L.L., and Salmon, L.G., 1998, "Introduction of TwoLink, InPlane Bistable Compliant ME MS," Proceedings of the 1998 ASME Design Engineering Technical Conferences, DETC98/MECH5837. [19] Jensen, B.D., "Identification of Macroand MicroCompliant 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 PseudoRigidBody Model," Microelectomechanical Systems (MEMS) 1997, presented at the 1997 ASME International Mechanical Engineering Congress and Exposition, November 1621, 1997, Dallas, Texas, DSCVol. 62, pp. 119126. [21] SHIGLEY, J.E. WICKER, J.J. Theory of machines & mechanisms McGrawHill, 1980.
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52[22] B.A. Coulter and R.E. Miller, Numerical analysis of a generalized plane elastica with nonlinear material behavior, Int J Numer Meth Eng 26 1988, pp. 617Â–630. [23] Howell, L.L. and Midha, A., 1994, "A Method for the Design of Compliant Mechanisms with SmallLength Flexural Pivots," ASME Journal of Mechanical Design, Vol. 116, No. 1, pp. 280290. [24] Howell, L.L., and Midha, A., 1996, "A LoopClosure Theory for the Analysis and Synthesis of Compliant Mechanisms," ASME Journal of Mechanical Design, Vol. 118, No. 1, pp. 121125. [25] Salmon, L.G., Gunyan, D.B., Der derian, J.M., Opdahl, P.G., and Howell, L.L., 1996, "Use of the PseudoRig id Body Model to Simplify the Description of Compliant MicroMechanisms," 1996 IEEE SolidState and Actuator Workshop Hilton Head Island, SC, pp. 136139. [26] Salamon, B.A., and Midha, A., 1992, "An Introduction to Mechanical Advantage in Compliant Mechanisms," Advances in Design Automation, (Ed.: D.A. Hoeltzel), DEVol 442, 18th ASME Design Automation Conference, pp. 4751. [27] Howell, L.L., Midha, A ., and Norton, T.W., 1996, "E valuation of Equivalent Spring Stiffness for Use in a PseudoRigidBody Model of LargeDeflection Compliant Mechanisms," ASME Journal of Mechanical Design, Vol. 118, No. 1, pp. 126131. [28] M. R. Spiegel and J. Liu, Schaum's Outlines: Mathematical Handbook of Formulas and Tables. New York, NY: McGrawHill, 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.
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53 Appendices
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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 Nonlinear Deflection /PREP7 !LCLEAR, ALL !LDELE, ALL !KDELE, ALL R=313.38 length in micrometers PI=acos(1.) h1=2 b1=20
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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
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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 #, XCo ord, YCoord, ZCoord) ********* 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
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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 12 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 34 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 56 !******* 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
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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 largedeflection e ffects in a static or full transient analysis
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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
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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
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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
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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
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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 Nonlinear Deflection
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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
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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
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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 #, XCo ord, YCoord, ZCoord) ********* 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 12
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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 56 !******* 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
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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 largedeflection 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
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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
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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
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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
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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
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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 Nonlinear 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 **************
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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 #, XCo ord, YCoord, ZCoord) ********* 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
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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 12 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 34 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 56 !******* 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
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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 largedeflection e ffects in a static or full transient analysis !AUTOTS, ON !CNVTOL,U,,0.000001,,0
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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
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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
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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
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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
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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 1.74532925E02 0.0000000 223.33658 3.49065850E02 0.0000000 446.66746 5.23598776E02 0.0000000 669.98624 6.98131701E02 0.0000000 893.28688 8.72664626E02 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
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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
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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
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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
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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
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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
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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
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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
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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 MFile The following file is an Mf ile that is used in MatLab. It reads the output of the ANSYS Batch files. MFile: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 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], '*' )
