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The Kinematic Analysis and Synthesis of Conic Constraint Pairs in Coplanar Motion

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The Kinematic Analysis and Synthesis of Conic Constraint Pairs in Coplanar Motion THE KINEMATIC ANALYSIS AND SYNTHESIS OF CONIC CONSTRAINT PAIRS IN COPLANAR MOTION By JACK WESLEY SPARKS A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1970 To Willie Wesley Sparks and Perry Lynn Sparks ACKNOWLEDGEMENTS The author expresses his appreciation to Dr. Delbert Tesar for the suggestion and supervision of this dissertation. The author is indebted to C.A . Morrison and J.A. Samuels for the financial support provided by their research contracts with the Southern Service Company and the Office of Civil Defense. Appreciation is given to the following committee members for their guidance and supervision, Dr. D. Tesar Dr. R.B. Gaither Dr. J. Mahig Dr. J. M. Vance Dr. R. G. Selfridge. Finally, the author extends his deepest appreciation to Ills father and mother, to his family and to his wife, Cheri , for their financial support and encouragement. 1 1 r TABLE OF CONTENTS Page ACKNOWLEDGEMENTS i i i LIST OF TABLES vii LIST OF FIGURES viii ABSTRACT . x CHAPTER I GENERAL BACKGROUND . 1 Conic Sections. 2 .Analysis of the Coupler Curve Function for Conic Constraint Pairs ... 4 Analysis of the Coupler Curve Function for Burmsster Constraints. ... 4 Linkage Synthesis for Burmester nd Conic Constraint Pairs 5 Kinematic Synthesis for Burmester Constraints 6 Kinematic Synthesis for Conic Constraints 6 The Application of Coplanar Synthesis . , 7 II BURMESTER THEORY THE STATE OF THE ART. .... 15 Burmester Constraints Variables Vs. Parameters. 17 Burmester Theory The Study of Five Multiply Separated Positions in Coplanar Motion 13 Nine Precision Point Synthesis for Burmester Constraints .... 32 IV Page III CONIC THEORY THE STATE OF THE ART 35 Conic Theory Variables Vs. Parameters ... 37 Six Position Conic Theory . 39 The Six Position Conic Section Point Curve 44 Verification of Circularity 47 The Describing Parameters of the General Conic 48 The Describing Parameters of the Parabola. ..... 51 Seven Position Conic Theory 56 IV ANALYSIS OF THE GENERALIZED CONIC CONSTRAINT PAIR 63 Higher Order Contacts . 67 Analysis of Burmester Constraints 69 Locating the Pole of the Moving Plane ... 72 Analysis of Arbitrary Points in the Moving Plane 7 3 Conversion to Real Time 74 V LINKAGE SYNTHESIS FOR THE GENERALIZED CONIC CONSTRAINT SET . 76 Linkage Synthesis for the Burmester Constraint Pair 79 Coupler and Fixed Link Synthesis for Conic Constraints 83 Coupler and Fixed Link Synthesis for Burmester Cons tx dints 86 VI SIX MULTIPLY SEPARATED POINT-POSITIONS IN COPLANAR MOTION 90 Page Algorithm or Six Multiply Separated Point-Positions in Coplanar Motion. VII NINE MULTIPLY SEPARATED PRECISION POINTS IN COPLANAR MOTION Synthesis cf Nine Symmetrical Precision Points 109 APPENDIX A. EXPANSION OF THE CIRCLEPOINT EQUATION BY DETERMINANTS .114 LIST OF REFERENCES. CITED 117 OTHER REFERENCES 119 BIOGRAPHICAL SKETCH 121 v 1 2- LIST OF TABLES 3- 3- TABlE Page 4- 1 MOTION COEFFICIENTS FOR THE BURMESTER CONSTRAINTS . 25 1 DESCRIBING COEFFICIENTS OF THE CONIC. 40 2 MOTION COEFFICIENTS FOR THE CONIC CONSTRAINTS 43 1 COEFFICIENTS OF ANALYSIS FOR THE CONIC - CONSTRAINTS ......... 70 VI LIST OF FIGURES FIGURES Page 1-1 Plane Intersecting a Cone in Space. 3 1-2 Posthole Digger Employing a Straight- Line Mechanism . 9 1-3 Function Generator (Decimal to Log Converter) 10 1-4 Dwell Mechanism 11 1-5 Conic Constraint Mechanism (Adjustable Step- Indexing Mechanism) . 12 1-6 Automatic Feed Mechanism Employing a Doub 1 e Dwe 1 1 C am 13 1-7 Geared Mechanism (Cycloidal Crank With Adjustable Amplifier) .... 14 2-1 Class I (Burmester Constraints) .... 16 2-2 Class II (Conic Constraints) ...... 16 2-3 Variables Vs . Parameters for Coplanar Synthesis 18 2-4 Translation to the Pole Reference . 20 2-5 Rotation and Scaling of the Fixed Reference System. .... 21 2-6 The Generalized Reference System. 23 2-7 Five Multiply Separated Positions for Case PPP-PP . 30 2-8 oint Synthesis toir Cnso PPP-P-P . 31 3-1 Typical Conic Constraints 3b 3-2 Properties of a Conic in the Fixed Plane . 38 viii 3- FIGURES Fage 4- 3 A Conic Constraint Pair Satisfying of . 4- Seven Positions the Moving Plane. 62 5- 1 Coupler Curve Function for Conic Constraints 66 5- 2 Coupler Curve Function for Burmester 6- Constraints 71 6- 1 Conic Set Synthesis While Satisfying Seven Precision Points 78 7- 2 Burmester Link Synthesis While Satisfying Five Precision Points .... 81 1 Six Multiply Separated Point-Positions in Coplanar Motion 92 2 Mechanism Satisfying Six Multiply Separated Point-Positions in Coplanar Motion 98 1 Deviation of a Burmester Set for Different Input 100 7-2 Mechanism Satisfying Nine Multiply Separated Precision Points in Coplanar Motion 108 7-3 Symmetrical Displacements of die Moving Plane 112 ix Abstract of Dissertation Presented to the Graduate Council in Partial Fulfilment of the Requirements for the Degree of Doctor of Philosophy THE KINEMATIC ANALYSIS AND SYNTHESIS OF CONIC CONSTRAINT PAIRS IN COPLANAR MOTION By Jack Wesley Sparks December, 1970 Chairman: Dr. Delbert Tesar Major Department: Mechanical Engineering A comprehensive study of the analysis and synthesis of conic constraint pairs in coplanar motion is presented in a generalized form. A state of the art is given for Burmester Theory and for the Generalized Conic Theory. A seven position algorithm for the Generalized Conic Theory provides two quartics relating the sixteen generalized co n i c po i n t s .. An analysis algorithm is shown to satisfy both, the Burmester mechanisms and the conic mechanisms and does not require iterative techniques for solution. The conversion to real time is shown to be an elementary transformation. Algorithms for prescribing fixed and moving pivots while satisfying five precision points provide extensions to the well-known graphical procedures of the past. These . algorithms are presented for both Burmester and Conic Theory An extension of Burmester Theory to the synthesis of six point-positions of the moving plane is presented with a theorem of uniqueness for six positions of the moving plane. The nine precision point problem is reduced to a solution of five equations with five unknown variables Tabulated coefficients for the analysis and synthesis algorithms provide a systematic and simplified means of computation for all case studies. xx CHAPTER I GENERAL BACKGROUND In 1967 the author completed research which coalesced the study of finite (positional) and infinitesimal (instantaneous) kinematics by presenting an algorithm for five multiply separated positions (any multiple combination of finite and/or infinitesimal positions) in coplanar motion. The work is similar to that outlined by Bottema [1] for the finite case and requires the resolution of four linear and two non-linear algebraic equations based on circular constraints in the fixed plane. A generalized computer program capable of solving all seven cases was the culmination of this work. Since its completion there have been few research publications associated with generalized positional synthesis for Burmester or Conic constraints. Consequently, as in many disciplines of study, kinematics appeared to have reached a plateau in its capabilities in coplanar synthesis. Often when this happens the only format to follow is to survey the field of discipline with new conceptual ideas to see if this might precipitate different concepts which will reflect upon and advance the particular area of study. Success is primarily dependent upon taking an interrogative viewpoint in the light of what is known and what has created such a . standstill. t This attitude generally allows one to gain new insight concerning his problem and most often establishes a solution. The technique employed in synthesizing six point- positions and nine precision points, presented herein, is a result of taking an interrogative look at coplanar synthesis and applies a new concept to obtain the resulting algorithms. The algorithms presented for six point- positions and for nine precision points are capable of solving all case studies for multiply separated point- positions in coplanar motion. Conic Sections It was Rene Descartes who in 1637 first applied algebra to geometry making the study of conic sections a part of elementary mathematics, As shown by Descartes, there are three general classes of curves obtained by the intersection of a plane and a cone as illustrated in Figure (1-1) and defined as follows: 1) An ellipse is formed wTien the plane section cuts only one nappe of a cone and is not parallel to an element of the cone. A circle is obtained as a special ellipse when the plane is perpendicular to the centerline and resolves to a point at the vertex 2) A parabola is obtained as a plane section becomes parallel to an element of the cone. A line is formed as a limiting case. 3) A hyperbola is formed in both nappes of the cone. Two intersecting lines are obtained as a limiting case. 3 TYPES OF CONICS 1) ELLIPSE 2 ) PARABOLA 3) HYPERBOLA 4} CIRCLE (SPECIAL ELLIPSE) Figure (.1-1) Plane Intersecting a Cone in Space 4 Analysis of the Coupler Curve Function for Conic Constrain t Pa irs In recent years many kinematic researchers have expended considerable effort to define and analyze the output function (coupler curve) of the simple four -bar mechanism. Since this study is concerned with conic constraint pairs in motion, an algorithm is presented to coalesce and generalize the analysis of the conic mechanisms with the mode of synthesis. A.s will be shown in the text, the analysis of the circular constraint system employs the same algorithm as the generalized conic constraint system. However, it becomes advantageous to discuss the two systems separately because of the difference in the complexity of the solutions The basic concern for the development of an analysis algorithm was to acquire a method of verifying the results provided by the synthesis program.
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