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Unified Higher Order Path and Envelope Curvature Theories in Kinematics

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Unified Higher Order Path and Envelope Curvature Theories in Kinematics UNIFIED HIGHER ORDER PATH AND ENVELOPE CURVATURE THEORIES IN KINEMATICS By MALLADI NARASH1HA SIDDHANTY I; Bachelor of Engineering Osmania University Hyderabad, India 1965 Master of Technology Indian Institute of Technology Madras, India 1969 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY December, 1979 UNIFIED HIGHER ORDER PATH AND ENVELOPE CURVATURE THEORIES IN KINEMATICS Thesis Approved: Thesis Adviser J ~~(}W>p~ Dean of the Graduate College To Mukunda, Ananta, and the Mothers ACKNOWLEDGMENTS I take this opportunity to express my deep sense of gratitude and thanks to Professor A. H. Soni, chairman of my doctoral committee, for his keen interest in introducing me to various aspects of research in kinematics and his continued guidance with proper financial support. I thank the "Soni Family" for many warm dinners and hospitality. I am thankful to Professors L. J. Fila, P. F. Duvall, J. P. Chandler, and T. E. Blejwas for serving on my committee and for their advice and encouragement. Thanks are also due to Professor F. Freudenstein of Columbia University for serving on my committee and for many encouraging discussions on the subject. I thank Magnetic Peripherals, Inc., Oklahoma City, Oklahoma, my em­ ployer since the fall of 1977, for the computer facilities and the tui­ tion fee reimbursement. I thank all my fellow graduate students and friends at Stillwater and Oklahoma City for their warm friendship and discussions that made my studies in Oklahoma really rewarding. I remember and thank my former professors, B. V. A. Rao, V. Ramamurty, and K. Lakshminarayana of IIT Madras, India, for their advice and encouragement. I lovingly acknowledge my mother, Seshagiramma, for her affection, untiring eiiorts, and encouragement that made my higher education a definite possibility. My affections are due to my wife, Dr. Visalakshmi, for her love and encouragement. iii TABLE OF CONTENTS Chapter Page I. INTRODUCTION • 1 1.1 Path Curvature Theory 1 1.2 Technical Discussion • 4 1.3 Intrinsic and Instantaneous Invariants 7 1.4 Tangent Line Envelope Curvature Theory 8 1. 5 Thesis Organization . • . • . • . • . 8 II. INTRINSIC AND INSTANTANEOUS INVARIANTS 10 2.1 First Order Motion ..•. 11 2.1.1 Rotation •• 11 2.1.2 Translation- 13 2.2 Location of S1 . 13 2.3 Second Order Motion 14 2.3.1 Rotation 14 2.3.2 Translation • • . 16 2.4 Location of Point S on S1 17 2.5 Third Order Motion . 18 2.5.1 Rotation . 19 2.5.2 Translation 21 2.6 Instantaneous Invariants 23 2.7 Numerical Example 25 2.8 Determination of Rigid Body Motion From Com­ ponents of Point Translations in the Body 28 2.8.1 First Order Motion . 28 2.8.2 Second Order Motion 30 2.8.3 Third Order Motion 30 III. PROPERTIES OF POINT-PATHS 32 3.1 First and Second Order Properties of Point-Paths 32 3.2 Inflection Points 33 3.3 Alternative Set of Equations for the Inflection Curve 36 3.4 Numerical Example 37 3.5 Tbird Order Properties . 39 3.6 Ball Points 40 IV. STRETCH ROTATIONS 41 4.1 Order of Contact and Stretch-Rotation of a Point-Path . • • • • • . • • • • . • • • • • • 41 iv Chapter Page 4.1.1 Stretch-Rotation Characteristic Numbers 43 V. CHARACTERISTIC EQUATIONS 45 5.1 Al-Equation . 45 5.2 >..1-Equation in Spherical Kinematics . 47 5.3 >..1-Equation in Plane Kinematics . 48 5.4 S1-Equation . 49 5.5 SrEquation in Spherical Kinematics . 50 5.6 Intersections of A.l and sl Surfaces and Cones . 51 5.7 Numerical Examples . 53 VI. PLANAR ENVELOPE CURVATURE THEORY 57 6.1 Canonical System and Instantaneous Invariants for Rigid Body Motion • . • • • • • . • • • 57 6.2 Instantaneous Invariants for the Tangent Line 59 6.3 Equation of the Envelopes . • • • • • 61 6.4 Envelopes Having the Same Radius of Curvature • 61 6.5 Envelopes ~Vi th the Same Third Order Characteristic 63 6.6 Envelopes With the Same Third and Fourth Order Characteristics 65 VII. SUMI·1ARY AND SCOPE FOR FURTHER STUDIES 69 7.1 Summary • • . • • • . 69 7.2 Significant Contributions 71 7.3 Scope for Further Studies . 72 7.3.1 Fourth Order Path Curvature Theory 72 7.3.2 Industrial Applications of Envelope Curvature Theory . • • . • • • 72 7.3.3 Kinematics of Higher-Order Tangent­ Plane Space-Envelope Curvature Theory • • • • • • • 72 7.3.4 A New Method to Study the Curvature Properties of Ruled Surfaces Generated by Space Mechanisms • • • • 76 BIBLIOGRAPHY 79 v LIST OF TABLES Table Page I. State-of-the-Art in Research in Higher Order Curvature Theory • . • 5 II. Instantaneous Invariants for Total Motion 24 III. Instantaneous Invariants for the Coupler 27 IV. Points on Inflection Curve 38 v. Points on the A1-l\ Curve . 54 VI. Points on the >..1-sl Curve 55 VII. Points on the >..1-f\ Curve . 56 vi LIST OF FIGURES Figure Page 1. First Order Spatial Motion of a Rigid Body • • 12 2. Second Order Spatial Motion of a Rigid Body 15 3. Third Order Spatial Motion of a Rigid Body 20 4. RCSR Spatial ~1echanism • . 26 5. Measurement Locations for Velocity, Acceleration and Super Acceleration in a Hoving Body in Space . 29 6. Stretch Rotation of Curves 42 7. Tangent Line LL Lying in a Rigid Body L 60 8. Locus of Tangent Lines Generating Envelopes With Equal Radius of Curvature • . • . • • • 64 9. A1 and A2 Curves for a Four-Bar Mechanism 66 10. Location of a Tangent Plane u Using a Vector OP Nor~al to the Plane • . 73 11. Example of a Developable Surface and Its Relationship to the Edge of Regression . • • 75 12. Locating a Line in Space . 77 vii CHAPTER I INTRODUCTION Synthesis of mechanisms that produce constrained mechanical motion with reasonable accuracy has been one of the concerns of engineers ever since man started harnessing nature. Reuleaux [1], author of one of the earliest classical textbooks on kinematics, observed a century ago that our success in synthesis would be proportional to the work spent in analysis. A recent survey by Soni and Harrisberger [2] and the biblio­ graphies [3, 4] on the development of mechanisms literature prove the truth of the statement of Reuleaux. The present study is concerned with the higher order motion analysis of a rigid body and points and lines in that body. This analysis is directly applied in the kinematic synthesis of mechanisms by way of determining points and lines in coupler links to generate desired paths and envelopes. 1.1 Path Curvature Theory Curvature theory [l-14] has been a fascinating subject of investiga­ tion to both mathematicians and kinematicians ever since L'Hospital (1696) treated the problem of path and envelope curvature. Problems of curva­ ture theory related to the presently known Euler-Savary Equation were studied by a number of mathematicians during the eighteenth and nine­ teenth centuries, and the contributions have added many dimensions to this subject. De La Hire (1706), for example, discussed the existence of 1 2 the inflection circle using a geometric approach. Bresse (1853) approach- ed the same problem using kinematic considerations. Euler (1765) exam- ined the problem of envelopes in the case of circular centrodes, and Savary (1831) made further studies on curvature, presenting a number of geometric constructions applicable for a variety of design situations. The relationship governing center of curvature for the path of any point of a moving plane and the known paths of other points came to be known as the Euler-Savary Equation. Later developments have been in the deter- ruination of points in a body that generate circular and straight paths up to the third or fourth order. Freudenstein [15] brought curvature theory into the limelight again in recent years when he published his landmark paper on generalized cur- vature theory introducing the concept of stretch rotation. He introduced (n-2) characteristic numbers to describe curvature properties of a plane curve within stretch rotations up to the nth order. Applying this con- cept of stretch rotation in analysis and synthesis of planar mechanisms, characteristic numbers A1 and A2 are defined respectively for the first and second rates of change of path curvature in terms of the evolutes to the given curve. Characteristic equations for the locus of Al and A2 points were derived and applied to solve various synthesis problems. While Freudenstein treated his work in polar coordinates, Veldkamp [16] introduced rectangular coordinates and remarked on ball points and T-positions, which he discussed extensively in his earlier work on curva- ture theory [14] using instantaneous invariants first introduced by ) Bottema [17]. Roth and Yang [35], Gupta [34], and Soni, Siddhanty and Ting [40] extended the study and application of these invariants in plane kinematics. 3 Kamphuis [18] extended the concepts of instantaneous invariants for the study of circular higher order curvature theory in spherical motion. This work has been generalized by Yang and Roth [19] on higher order path curvature in spherical kinematics in similar lines as Freudenstein's. They defined stretch rotation characteristic numbers A's for the curva­ tures of a spherical path and obtained characteristic equations for the locii of points having the desired A-path characteristics. Skreiner [20, 21] made a thorough study of geometry and kinematics of instantaneous spatial motion using elementary methods in vector alge­ bra. He investigated special surfaces in a moving body characterized by one or the other property of tangential and normal components of accel­ eration of points contained in them and derived expressions for a family of ellipsoids characterized by magnitudes of total accelerations in the moving body. His contribution includes the study of inflection points in spatial motion.
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