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Kinematic Synthesis and Analysis Techniques to Improve Planar Rigid-Body Guidance

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Kinematic Synthesis and Analysis Techniques to Improve Planar Rigid-Body Guidance KINEMATIC SYNTHESIS AND ANALYSIS TECHNIQUES TO IMPROVE PLANAR RIGID-BODY GUIDANCE Dissertation Submitted to The School of Engineering of the UNIVERSITY OF DAYTON In Partial Fulfillment of the Requirements for The Degree Doctor of Philosophy in Mechanical Engineering by David H. Myszka UNIVERSITY OF DAYTON Dayton, Ohio August 2009 KINEMATIC SYNTHESIS AND ANALYSIS TECHNIQUES TO IMPROVE PLANAR RIGID-BODY GUIDANCE APPROVED BY: Andrew P. Murray, Ph.D. James P. Schmiedeler, Ph.D. Advisor Committee Chairman Committee Member Associate Professor, Dept. of Associate Professor, Dept. of Mechanical and Aerospace Engineering Mechanical Engineering, University of Notre Dame Ahmad Reza Kashani, Ph.D. Raul´ Ordo´nez,˜ Ph.D. Committee Member Committee Member Professor, Dept. of Mechanical and Associate Professor, Dept. of Electrical Aerospace Engineering and Computer Engineering Malcolm W. Daniels, Ph.D. Joseph E. Saliba, Ph.D. Associate Dean of Graduate Studies Dean School of Engineering School of Engineering ii c Copyright by David H. Myszka All rights reserved 2009 ABSTRACT KINEMATIC SYNTHESIS AND ANALYSIS TECHNIQUES TO IMPROVE PLANAR RIGID- BODY GUIDANCE Name: Myszka, David H. University of Dayton Advisor: Dr. Andrew P. Murray Machine designers frequently need to conceive of a device that exhibits particular motion char- acteristics. Rigid-body guidance refers to the design task that seeks a machine with the capacity to place a link (or part) in a set of prescribed positions. An emerging research area in need of advancements in these guidance techniques is rigid-body, shape-change mechanisms. This dissertation presents several synthesis and analysis methods that extend the established techniques for rigid-body guidance. Determining link lengths to achieve prescribed task positions is a classic problem. As the number of task positions increases, the solution space becomes very limited. Special arrangements of four and five task positions were discovered where the introduction of prismatic joints is achievable. Once dimensions of link lengths are determined, solutions with defects must be eliminated. De- fects include linkages that do not remain on the same assembly circuit, cross branch points where the mechanism cannot be driven, or achieve the task positions in the wrong order. To address the circuit defect, a general strategy was formulated that determines the number of assembly config- urations that exist in a linkage and the sensitivity of the motion to a link length. To address the branch defect, extensible equations that generate a concise expression of the singularity conditions iii of mechanisms containing a deformable closed loop was developed. To address the order defect, an intuitive method to ensure that rigid-bodies will reach design positions in the proper order has been developed. The result of the dissertation is a suite of methodologies useful in the synthesis and analysis of planar mechanisms to accomplish rigid-body guidance. iv To Kim and our children: Jordan, Taylor, Payton and Riley v ACKNOWLEDGMENTS The writing of this dissertation was a tremendous undertaking and not possible without the personal and practical support of several people. To that point, my sincere gratitude goes to my family, my advisor and my committee members for their support and patience over the last few years. First, I thank my advisor Dr. Andrew Murray, for his encouragement, guidance, and friendship. I am grateful that Drew took a very active approach throughout my research. He showed me how to strive for the underlying explanation for observations encountered in research work. He taught me different ways to approach a problem and the need to be persistent to accomplish an objective. I greatly admire his passion for kinematics, mathematical savvy and commitment to his students. I also wish to express appreciation to Dr. James Schmeideler for his guidance and careful review of several concepts, theories and writings. His mastery of kinematic principles and foresight into possibilities is remarkable. I admire his knowledge, insightfulness and dedication to his profession. Besides my advisors, I would like to thank the other members of my doctoral committee: Dr. Raul´ Ordo´nez˜ and Dr. Reza Kashani who asked stimulating questions and offered useful input. Finally, it would have been impossible to pursue my research goal without my family’s love and support. This dissertation is dedicated to them. vi TABLE OF CONTENTS Page ABSTRACT . iii DEDICATION . v ACKNOWLEDGMENTS . vi LIST OF FIGURES . xi LIST OF TABLES . xvi CHAPTERS: I. INTRODUCTION . 1 1.1 Shape Changing, Rigid-Body Mechanisms . 2 1.2 General Synthesis Concepts . 4 1.2.1 Two-Position Guidance . 9 1.2.2 Three-Position Guidance . 10 1.2.3 Four-Position Guidance . 10 1.2.4 Five-Position Guidance . 11 1.2.5 Greater than Five-Position Guidance . 11 1.3 Solution Rectification . 11 1.3.1 Circuit Defects . 12 1.3.2 Branch Defects . 13 1.3.3 Order Defects . 14 1.4 Objectives . 15 1.5 Organization . 16 II. POSITION AND POLE ARRANGEMENTS THAT INTRODUCE PRISMATIC JOINTS INTO THE DESIGN SPACE OF FOUR AND FIVE POSITION RIGID-BODY SYN- THESIS . 19 2.1 Introduction . 19 2.2 Four Position Cases . 21 vii 2.2.1 Poles, the Compatibility Linkage, and Center-point Curves . 21 2.2.2 Transition Linkage . 24 2.2.3 Opposite Pole Quadrilateral with T1 = T2 = T3 = 0: A Rhombus . 25 2.2.4 Opposite Pole Quadrilateral with T1 = T2 = 0: A Parallelogram . 28 2.2.5 Opposite Pole Quadrilateral with T1 = T3 = 0 or T2 = T3 = 0: A Kite . 31 2.2.6 Opposite Pole Quadrilateral with T1 = 0 or T2 = 0 or T3 = 0 . 33 2.3 Synthesizing PR Dyads . 35 2.4 Synthesizing RP Dyads . 40 2.5 Three Position Case . 43 2.5.1 Collinear Poles . 44 2.5.2 Equilateral Triangle . 46 2.6 Five Position Cases . 46 2.6.1 Case 1: Movable Compatibility Platform . 47 2.6.2 Case 2: Alternate Configuration of the Movable Compatibility Platform . 48 2.7 Example 2.1 . 53 2.8 Summary . 53 III. USING A SINGULARITY LOCUS TO EXHIBIT THE NUMBER OF GEOMETRIC INVERSIONS, TRANSITIONS, AND CIRCUITS OF A LINKAGE . 57 3.1 Introduction . 57 3.2 Loop Closure Equations . 60 3.2.1 Input/Output Function . 61 3.3 Singularity Function . 63 3.4 Projection of the Singularity Locus . 66 3.5 Circuits . 68 3.6 Transition Functions . 70 3.7 Motion Possibilities . 71 3.8 Examples . 75 3.8.1 Example 3.1: Slider-Crank Mechanism . 75 3.8.2 Example 3.2: Watt II Mechanism . 78 3.8.3 Example 3.3: Stephenson III Mechanism, Actuated through the Four-Bar Loop . 79 3.8.4 Example 3.4: Stephenson III Mechanism, Actuated through the Five-Bar Loop . 85 3.9 Summary . 87 IV. SINGULARITY ANALYSIS OF AN EXTENSIBLE KINEMATIC ARCHITECTURE: ASSUR CLASS N, ORDER N − 1 ........................... 90 4.1 Introduction . 90 viii 4.2 Assur Class Mechanisms . 91 4.3 Singularity Analysis of a Mechanism with Three Links in the Closed Contour . 93 4.4 Singularity Analysis of a Mechanism with a Four-Link Deformable Closed Con- tour: Assur IV/3 . 96 4.5 Singularity Analysis of a Mechanism with a Five-Link Deformable Closed Con- tour: Assur V/4 . 100 4.6 Singularity Analysis of a Mechanism with an N-Link Deformable Closed Con- tour: Assur N/(N − 1) ............................... 102 4.7 Singularity Equation of a Mechanism with a Six-Link Deformable Closed Con- tour: Assur VI/5 . 104 4.8 Singularity Equation of a Mechanism with a Seven-Link Deformable Closed Contour: Assur VII/6 . 105 4.9 Examples . 108 4.9.1 Example 4.1: Combination of physical and joint parameters resulting in a singularity . 108 4.9.2 Example 4.2: Singularity with adjacent parallel links . 109 4.9.3 Example 4.3: Singularity with isolated parallel links . 110 4.10 Summary . 112 V. ASSESSING POSITION ORDER IN RIGID BODY GUIDANCE: AN INTUITIVE APPROACH TO FIXED PIVOT SELECTION . 114 5.1 Introduction . 114 5.2 Ordered Dyads . 115 5.3 Center-Point Theorem . 115 5.4 Necessary Pole Condition . 118 5.5 Sufficient Pole Condition . 120 5.6 Propeller Theorem . 121 5.7 Examples . 123 5.7.1 Example 5-1 . 124 5.7.2 Example 5-2 . 124 5.7.3 Example 5-3 . 128 5.7.4 Example 5-4 . 128 5.8 Summary . 130 VI. CONCLUSIONS AND FUTURE WORK . 132 6.1 Contributions of the Dissertation . 132 6.2 Directions for Future Work . 134 ix APPENDICES: A. SLIDER CRANKS AS COMPATIBILITY LINKAGES FOR PARAMETERIZING CEN- TER POINT CURVES . ..
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