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3/M/65 B an ANALYSIS of a SPATIAL FOUR-BAB In presenting the dissertation as a partial fulfillment of the requirements for an advanced degree from ";he Georgia Institute of Technology^. I agree that t2ie Library of the Institute shall make it available for inspection and circulation in accordance with its regulations governing materials of this type. I agree that permission to copy from, or to publish from, this dissertation may be granted by the professor under whose direction it was written, or, in his absence, by the Dean of the Graduate Division when such copying or publication is solely for scholarly purposes and does not involve potential financial gain. It is under­ stood that any copying from, or publication of, this dis­ sertation which involves potential financial gain will not be allowed without written permission. ^—7 rr 3/M/65 b AN ANALYSIS OF A SPATIAL FOUR-BAB. LINKAGE BY THE METHOD OF GENERATED SURFACES A THESIS Presented to The Faculty of the Graduate Division by Harry Price Gray In Partial Fulfillnient of the Requirements for the Degree Master of Science in Mechanical Engineering Georgia Institute of Technology May, I968 AN ANALYSIS OF A SPATIAL FOUR-MR LINKAGE BY THE lvffiTHOD OF GENERATED SURFACES Approvedi —— . / Chairman "~~^~Z^_____^2__ZAJL Date approved by Chairman: ii ACKNOWLEDGMENTS The author wishes to express sincere appreciation to all of those who have assisted in making this work possible. Special thanks are extended to the author's thesis advisor. Dr. F. R, E. Crossley, for his encouragement and assistance, and also to Dr. H. L« Johnson and Dr. A. W. Harris for serving on the thesis advisory committee, Thanks are also due to the National Science Foundation for its financial assistance in the forms of a graduate traineeship and NSF Grant Number GK-1203, This work is dedicated to the author's parents. iii TABLE OF CONTENTS Page ACKNOWLEDGMENTS ii LIST OF TABLES . v LIST OF ILLUSTRATIONS vi ABSTRACT viii NOMENCLATURE ix Chapter I. INTRODUCTION ,,,•..,<,..,•«.. l Historical Background Mechanical Background II. METHOD OF GENERATED SURFACES ......,..*•.... 7 Occurrence of Surfaces Application of Surfaces to Linkages III. PROBLEM INTRODUCTION .,.....«•„..*...... 16 The RCGR Mechanism and its Related Surfaces Equations of Torus and Cylinder IV. TORUS-CYLINDER INTERSECTIONS ...... .„0 .... 2^ Survey of Intersections Dimensional Restrictions V. INPUT-OUTPUT RELATIONS ................. 38 Intersection Equation and Solution Illustrated Results VI. CONCLUSIONS 'CABLE OF CONTENTS (Cont.) Page APPENDICES A. DERIVATION OF TORUS AND CYLINDER EQUATIONS ,.•..., 60 B. COMPUTER PROGRAM « 6h BIBLIOGRAPHY , . 71 LIST OF TABLES Surfaces Generated by Single Pairs and Two-Pair Combinations Forming an Open Chain (Excluding the Screw Pair). LIST OF ILLUSTRATIONS Figure 1-a Revolute (R) Pair „,.,., o * o o a 1-b Prismatic (p) Pair 1-c Screw (s) Pair •#..-., o e a « o 1-d Cylindric (C) Pair ...,.• » • » • 1-e Ball or Globe (G) Pair • • a 1-f Planar (F) Pair . •..•.*. 2-a Generation of Right Circular Cylinder by a (C) Pair 2-b Generation of Sphere by a (G) Pair . „ „ „ . „ 2-c Generation of Plane by a (P) Pair „ „ , 3 The RCGR Mechanism .......„.....,,,, k General Orientation of Torus and Cylinder „ . „ a 5 Aid for Defining Mechanism - Surface Relations , , 6-a One-loop Intersection - Case a , 6-b One-loop Intersection - Case b , 6-c One-loop Intersection - Case c . 7-a Two-loop Intersection - Case a 7-b Two-loop Intersection - Case b , 7-c Two-loop Intersection - Case c . 7-b Two-loop Intersection - Case d , , 8-a Three-loop Intersection - Case a , 8-b Three-loop Intersection - Case b . LIST OF ILLUSTRATIONS (Cont,) Figure Page 8-c Three-loop Intersection - Case c . , , • . 32 8-d Three-loop Intersection - Case d . ......... • , 32 9-a Four-loop Intersection - Case a ............ 33 9-b Four-loop Intersection - Case b ...... * . 33 9-c Four-loop Intersection - Case c .„„,.....„.„ 34 10 Two-loop Intersection ,..,........,,... 3J4 11-a Projection of Torus and Cylinder in xy Plane . , . , . 37 L-b Illustration Defining r and r ............ 37 12 Projection of Torus Generator in xy Plane ....... ko 13-a Angle 0 versus Angle Y - Case #1 ........... 1+6 13-b Angle 3 versus Angle f - Case #1 .... ....... ^7 13-c Length d versus Angle Y - Case #1 .......... 4? l4-a Angle 3 versus Angle Y - Case #2 ........... „ i|8 Xk-lb Length d versus Angle Y - Case #2 . .... .... ^9 1^-c Angle 0 versus Angle Y - Case #2 ........ 0 » • M-9 15-a Angle 0 versus Angle f - Case #3 ....... .... $0 15-b Length d versus Angle f - Case #3 ........ 51 15-c Angle 3 versus Angle Y - Case #3 ..... ..... 52 l6-a Length d versus Angle ¥ - Case #4 .......... 5U l6-b Angle p versus Angle Y - Case #4 ........... 5^ 17-a Length d versus Angle Y - Case #5 55 17-b Angle 3 versus Angle Y - Case #5 . 55 l8-a Angle 3 versus Angle Y - Case #6 5^ l8-b Length d versus Angle Y - Case #6 57 viii ABSTRACT The characteristic motions of all RCGR mechanisms are examined using the method of generated surfaces, which is a method recently developed for studying spatial mechanisms• It is shown that the motions of this linkage can he represented hy the intersection of a cylinder and a torus, having any size and relative positions. The general character of these various intersections are first studied graphically and intuitive­ ly. Then a digital computer program is written by which the stepwise small displacements of any representative linkage can be calculated and tabulated. NOMENCLATURE a revolute pair a prismatic pair a cylindric pair a global pair a screw pair a planar pair mobility number of linkages number of "unfreedoms" at a joint a kinematic pair joining link #i and link # j an arbitrary point in a link in a kinematic chain the common perpendicular between two joint axes the angle between the cylinder axis and the z axiss measured counter clockwise the radius of the cylinder the radius of the generating circle, or the secondary radius of the torus the length of the common perpendicular between the cylinder axis and the z axis through the center of the torus the height above the xy plane of the common perpendicular- be tween the cylinder axis and the z axis the angle of rotation of the cylinder radius the angle of rotation of the secondary radius of the torus the angle of rotation of the primary radius of the toruss or the input angle NOMENCLATURE (Cont.) the angle of inclination of the plane of the generating circle about p, away from the vertical plane the angle of inclination of the plane of the generating circle about a line through its center and parallel to the z axisj away from the plane containing p and the z axis the primary radius of the torus 1 CHAPTER I INTRODUCTION Historical Background All real mechanisms are three-dimensional. For theoretical exami­ nation, however, they are usually divided into two categories — spatial mechanisms and planar mechanisms. The axes of the joints connecting the bodies, or links, of a planar mechanism must be parallel; and the motion of all links, relative to a fixed link, must be in a single plane or in parallel planes. When at least two joint axes are skew to one another and motion is in three dimensions, the mechanism belongs to the spatial class. Spherical mechanisms are special members of the spatial category which have all joint axes intersecting in a single point; and the links move on a spherical surface with that point of intersection as its center. Spatial mechanisms are less restricted as to possible configurations than are planar types, It follows that they can provide a greater variety of motion. However, the lack of restriction also makes the analysis and synthesis of spatial mechanisms much more difficult. Since planar mechanisms have motion only in parallel planes, their positions^ displace­ ments, and velpcities can be projected into a single plane; and concepts from plane geometry can be applied. Necessary mathematical calculations are relatively simple. The general spatial mechanism does not lend it­ self to such a direct approach, and computations are complicated, This complexity of design problems has hindered the practical involved planar mechanisms are now being used where simpler spatial types could perform the same task with increased accuracy and economy. For example, a simple three-bar spatial mechanism can provide a trans- lational output perpendicular to the plane of rotational output, while no planar form can have such a combination of motion. Recently, there has been increased activity in the study of spa­ tial mechanisms with the aim of making them more available for design use. Graphical and mathematical techniques have been devised that can be used successfully for the analysis and synthesis of a general spatial linkage. A. T. Yang (9) has worked with dual numbers and quaternions, Hartenberg and Denavit (l) have shown that matrices of "pair variables" can be applied to the synthesis of spatial mechanisms, and K. H. Hunt (2) uses instantaneous screw axes as an aid to analysis. These are just a few of the recent contributions made in this field. However, there is still a need for the development of design methods and aids that can be employed easily, thus making the utilization of spatial mechanisms more attractive. Mechanical Background The bodies that form a linkage are connected by joints which allow the bodies to have motion relative to one another. These joints, called kinematic pairs, are simply the contact points between adjacent links and can be classified, in a broad sense, by the type of contact. Lower pairs are joints which have surface contact, i.e. the two con­ nected members have surface areas touching. Higher pairs are joints 3 at which the two adjacent links have only line or point contact.
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