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. This paper will be concerned solely with lower pairs,
To locate the position of a body or link in space, six variables must be designated. Three define the translation of a point (along the x, y, and z axes, for instance), and three give the rotation of the body around that point (about the three coordinate axes). A single body in space, then, is said to have six degrees of freedom since there are six position parameters which can be "varied independently, according to some reference system. (Rectangular coordinates - x, y, and z - will be used as the reference system in this paper).
When a body is joined to another body, its motion relative to the first is constrained to some extent. The number of degrees of relative freedom present is defined by the number of variables needed to describe its motion relative to the first; and this quantity depends upon the kind of joint used to connect the two bodies.
The relative motion allowed between two links also depends upon the pair joining them. The broad category of lower pairs can now "be sub-classified by the motion and degree of relative freedom that each permits. Using the nomenclature of Hartenberg and Denavit (l), there are six lower pairs:
(l) The revolute pair (Figure 1-a), designated R, permits rota tion about a single axis. As shown in the figure, the motion of link #2 relative to link #1 can be described by one variable, angle 8 (measured from an arbitrary initial position), Tims the revolute pair has one degree of relative freedom. k
(2) The prismatic pair (Figure 1-b), designated P, allows only
translation along a single axis. Since the relative motion can be de
scribed by the single variable, x (measured from an arbitrary initial
position), the prismatic pair has one degree of relative freedom.
(3) If body #2 is allowed to rotate around and translate along
an axis with respect to body #1, the pair is cylindric. Figure 1-c
shows a cylindric pair, designated C. There are two degrees of rela
tive freedom.
(k) As its name implies, a screw pair (Figure 1-d), designated
S, allows a "helical" motion of body #2 relative to body #1. As body #2
rotates around the joint axis, it also has a definite translational dis
placement along the axis. Since the translation is specifically re
lated to the rotation by a constant (the pitch of the screw), there is
only one degree of relative freedom.
(5) A spheric joint, shown in Figure 1-e, eliminates all re
lative translationai motion between link #1 and link #2 without imped
ing the rotational freedoms. Thus link #2 is able to rotate about the
three coordinate axes with respect to link #1. The spheric joint
(sometimes called the ball-and-socket or global joint) has three degrees of relative freedom. Its symbol is G.
(6) Suppose the joint is formed by two flat planes in contact so that link #2 can slide on link #1 as shown in Figure 1-f. There are
three degrees of freedom between links #1 and #2 — one freedom of ro
tation and two freedoms of translation* The symbol for this pair is F.
It can be thought of as the limiting case of the G pair as the radius of the sphere approaches infinity. . Link #2 (' 1 x ^S> fi Link #1
Figure !• :.••-,) i..it i, if i.<;ir.-e l-b« Prismatic (P) Pair
Body // ••.. 0 Body #1 ;/ V;Wff/f X "tody /ft .>^ Body /
Figure 1-d. Cylindric (C) Pair Figure 1-c. Screw (S) Pair
Link #2 '• 1
X v83 // , '* 2 X r^L-Lii* #1 ,W,.//
Figure 1-c. Ball or Globe (G) Pair Figure 1-f. Planar (F) Pair
Figure 1, The Six lower Pairs [From Hartenberg and Denavit (l)] •:•
A four-bar linkage is composed of four bodies joined by four
kinematic pairs to form a single closed loop. Its mobility is defined
as a measure of its total degrees of freedom, that is, how many inputs
must be specified to define the motion of all the Units in the mechan
ism. If the mobility, M, is less than one, the linkage is a structure
and cannot move. If M is greater than or equal to one, then M inputs
are required to adequately define the motion of the mechanism. (M is
always an integer). The mobility of a mechanism was found by Kutzbach
to be defined, for the general case, by:
M - 6(n - 1) - Xu where M is the mobility, n is the number of linkages, and u is the
"unfreedom" of each kinematic pair. (The number of "unfreedoms" in a pair is six minus the degree of relative freedom of the pair) . Most mechanisms in use have a mobility of one, and this paper will be con
cerned only with mechanisms of that mobility. CHAPTER II
THE METHOD OF GENERATED SURFACES
Occurrence of Generated Surfaces
Let Q, be an arbitrary point in a link that is moving through all of its possible positions relative to another, fixed link. If, during this motion, point Q is able to become coincident with every point of a surface and that surface, in turn, contains all possible positions of
Q, then Q is said to "generate" that surface with respect to the fixed link. (Similarly, a point can, at other times, generate a curve or a volume),
A point Q can generate a surface relative to a fixed link when
Q has two degrees of freedom relative to the fixed link. Therefore, when a body (link #2) is joined to another body (link #l) by a kinematic pair that has two degrees of freedom, an arbitrary point, Q, in link #2 can generate a surface with respect to link #1, For example, consider the cylindric pair, which is the only lower pair with two degrees of freedom. In this, any point in body #2 will generate a right circular cylinder with respect to body #1, as shown in Figure 2-a.
If link #2 is joined to link #1 by a G pair, link #2 has actually three degrees of freedom relative to link #1, However, any point Q, which is in link #2 and therefore joined rigidly to the center of the spheric joint, can move with only two degrees of freedom because a rotation of link #2 abcut line QC will not change the position of Q relative to link #1, Point Q, therefore, can move over the surface of
a sphere concentric with the center of the spheric joint (Figure 2-b).
The F pair can be regarded as the limiting case of the spheric pair as the radius of tie spheric joint approaches infinity. Again body
#2 has three degrees of freedom relative to body //l. However., an arbi
trary point? Q, in body #2 has only two relative degrees of freedom since a rotation of boc}y #2 about a line through Q perpendicular to the plane of contact of the planar pair will not change the position of Q. Point
Q, then,generates a plane that is parallel to the plane of contact of the pair, as shown in Figure 2-c.
The F pair, the C pair, and the C pair are the only lower pairs that can form a two link open chain in which an arbitrary point in one link will generate a surface with respect to the other, (Note that lower pairs have the special property that the relative motion of the two links joined by the pair is the same regardless of which body is the reference link and which the moving link. This means that if a point, Q, is considered to be in body //2, moving relative to fixed body #1, the possible motion of Q is identical to that occurring when Q is considered to be in body #1 with body #2 fixed). It will now be shown that certain three-link, open chains may contain points that can generate surfaces.
Let ( ). . represent a kinematic pair joining link tf± and link # j«, I j
Then (R)1P represents a revolute pair joining link #1 and link #2.
Similarly, (R) — (C)0-, represents an open kinematic chain in which link
#2 is joined to link #1 by a revolute joint and link #3 is joined to link #2 by a cylindric joint. Right Circular Cylinde
#1
Figure 2-a. Generation of Right Circular Cylinder "by a (C) Pair (3)
Sphere
Body #1
Figure 2-b, Generation of Sphere by a (G) Pair (3)
Plane
Body #1
Figure 2-c, Generation of Plane by a (P) Pair 10
If three links are joined together ( ).?- ( )?^, they my pro vide a total of only two degrees of freedom between link #3 and link #1
if a surface is to be generated with respect to link #1 by an arbitrary point in link #3. Thus both ( ) and ( ) must be pairs with one degree of relative freedom. There exist only three lover pairs with one degree of freedom — the R pair, the P pair, and the S pair. Use of the screw pair results in surfaces that present unusual problems in analysis, and they will not be discussed here. Consequently, there
p remain only four combinations of two pairs: (R) — (R) , (R)-,p """ ( )?o)
(P)12 — (R)p?> and (P),p ~ (P)?.. The surfaces that can be generated from these combinations are cylinders, hyperboloids, tori, and degenerate or special forms of these surfaces. The special forms occur when the pair axes have a particular orientation rather than the general skewed arrangement.
Table 1 lists the lower pairs and combinations of two lower pairs that can generate surfaces, the surfaces they generate, and the special orientation of the axes required to produce certain surfaces. Link $1 always denotes the fixed link, and Q, is an arbitrary point in link #2 for an open chain of two links, or in link jf3 far an open chain of three links. The common perpendicular between two joint axes is designated by V.
For three-link open chains, the following five relative orienta tions of the joint axes need to be examined to identify all the forms of the generated surfaces. (These do not include cases where the surface degenerates to a curve or line), 11
Table 1, Surfaces Generated by Single Pairs and Two-Pair Combinations Forming an Open Chain. (Excluding the Screw Pair)
Kinematic Orientation of Pair Axes and V, Surface Generated Chain Common Perpendicular Between Axes by "Q"
(0 Right Circular 12 Cylinder (G)12 Sphere
(F)1£ Plane
(E)12 tH)23 Axes Skewed (General Case) Toru s - Ge ne ra t i ng Circle Skewed in Two Planes Axes Parallel Annular Plane
Axes Intersecting Sphere or Spheric Zone Axes are Perpendicular Right (Q chosen so perpendicular Circular from it to (R)?O axis Torus Intersects V) Axes are Perpendicular Torus-Generating Not Intersecting Circle Skewed in One Plane (5^7 (P)23 Axes Skewed (General Case) Hyperboloid Axes Intersecting Cone Not Perpendicular (O. is assumed to move along (P)23 Axis).
Axes Intersecting Plane and Perpendicular
Axes Parallel Right Circular Cylinder
Axes Perpendicular Plane and Two-Pair Combinations Forming an Open Chain. (Excluding the Screw Pair).
Kinematic Orientation of Pair Axes and V, Surface Generated Chain Comoron Perpendicular Between Axes by "(J11
(P)12(R)23 Axes Skewed (General Case) Cylinder Skewed in Two Planes
Axes Intersecting Cylinder-Skewed (Not Perpendicular) in One Plane
Axes Intersecting and Perpen Plane dicular
Axes Parallel Right Circular Cylinder
Axes Perpendicular Plane
(P) 2(P)2 All Cases Except "Axes Parallel" Plane (which results in degenerate form) 13
(1) The axes arq skewed. (This is the general case).
(2) The axes are intersecting'.
(3) The axes are intersecting and perpendicular,
(h) The axes are parallel. (5) The axes are perpendicular.
Application of Surfaces to Linkages
It has been shown that an arbitrary point in the "last" link of a two-link or three-link open chain can generate a surface with respect to the fixed link, (These surfaces have been listed in
Table l). Now these surfaces can be applied to the analysis of spatial mechanisms with three, four, or five links,
Suppose that there are two links, link #1 and link #2, which are each joined to the same reference link by lower pairs in such a way that both contain arbitrary points (Q! and Q") which have two degrees of freedom relative to the fixed link, (The possible single pairs and two-pair combinations that can give this result are listed on
Table l). Then both Q*, in link #1, and Q", in link #2, can each generate a surface relative to the fixed link. If the two surfaces intersect, then QT and Q," may be made coincident at all points of the intersection by joining them with some "bridge joint", and a single- loop mechanism will be formed, Q" and Q* must remain coincident, at a point 0, , while at the same time, each stays within its own surface,
Thus the possible paths of point 0, are given by the intersection of the two generated surfaces, It is desired that the mechanism have a mobility of one; and
therefore according to the Kutzbach criterion for single-loop mobility,
there must be a total of seven degrees of freedom among the kinematic
pairs. Since both Q' and Q" must have two degrees of freedom relative
to the fixed link, the joint connecting them must have three degrees of
freedom. Furthermore, Q' and Q" must be coincident, so no freedom may
be translational. Therefore, the bridge joint joining Q1 and Q" must
be a spheric pair (3).
The possible positions of the spheric pair, then, are defined
by the intersection of the two generated surfaces. If there is no
intersection, the spheric pair cannot be formed and the two generating
chains cannot be closed in a loop to form a linkage. If the intersection
is only a point (the two surfaces being tangent), the linkage can be
formed, but it cannot move. If the intersection exists, the curve of
intersection defines the possible path of the spheric joint; and since
the mobility of the mechanism is one, the corresponding motion of the
linkage can be determined.
The possible motion of these mechanisms depends upon the kinematic
pairs used, the arrangement of the pairs, and the ratios of the link
lengths. There are numerous questions that can be asked about the char
acteristic motion of a linkage formed from a given combination of pairs.
(l) Can the same linkage have more than one manner of assembly with the
same link-length ratios? (2) If so, how many? (3) Are there "fork"
points where a link in the mechanism can follow one of two intersecting paths, OT even three intersecting paths? (k) Can there be more than one of these points? (5) Is there an applicable three-dimensional Grashof Criterion? (6) Is there a maximum value for output oscillation with any input oscillation?
In this work, the method of generated surfaces will be applied to a particular family of linkages — those with two R pairs, a G pair, and a C pair - to investigate these questions. Other work with the intersection of surfaces has utilized graphical, vector, and analog computer techniques. In this paper, a digital computer is used to study the intersections, as well as intuitive and graphical methods. CHAPTER III
PROBLEM INTRODUCTION
The method of generated surfaces will now be applied to all
dimensional variations of the particular four-link spatial mechanism
which consists of two revolute joints, a cylindric joint, and a
spheric joint. One aim is to answer the questions about its character
istic motion that were presented at the end of Chapter II. This parti
cular combination of links was chosen because preliminary investigation
and intuition indicated that many different and interesting motions
may be produced by it.
There are two kinematic arrangements for the four given pairs;
these are (R)^- (C)^- (G)^- (R)^ and (R)^- (C)^- (R)^- (G)^.
The generated surfaces that correspond to ea:h of these combinations
must now be determined. This is done by disconnecting the (G) pair,
and then choosing a reference link such that ar arbitrary point on each of the two links that were joined by the (G) pair can generate a
surface relative to the reference link.
C G For the fa)^- ( ')23~ ( )^~ 00^ mechanism, link #3 and link #4 are separated by uncoupling the (G) pair. Then it might be
thought that either link #1 or link #2 be chosen as the reference link.
However, if link #1 be fixed, then the two generating chains are (R).. and (R) — (C)p_ and neither of these chains produce surfaces. (A point in link #h generates a circle with respect to link #1 •; a point in link #3 generates a volume with respect to link #1)• If, by con
trast, link #2 be selected as the fixed reference, the generating chains
are (C)?_ on the one side and (R) — (R) , on the other side. Both of these produce surfaces, as each has two degrees of freedom; Table 1 will verify this. An arbitrary point Q' in link #3 generates a right
circular cylinder with respect to link: #2, and an arbitrary point Q"
in link #4 generates a torus with respect to link #2.
R When the above procedure is applied to the (R) — (C)O-D~ ( )-3lT~
(G), mechanism, it is found that no appropriate reference link exists such that two open chains generate surfaces when the (G), pair is separated. Thus the method of generated surfaces cannot be applied to this arrangement; some other method must be used for analysis,
Since the method of generated surfaces can be applied to the
(R) — (C) — (G)olt~~ ^Vl arran6en]fin"t' an(i cannot be applied to the
(R) — (C)2 —(R) »— (G),_ arrangement, it can be predicted that the two have completely different characteristic motions. The movement of the spherical joint in the first set is described as following the inter section of two geometric surfaces — a cylinder and a torus? while the motion of the second is not. The (R) — (c)p^~ ^3ii"~ ^\i mechanism can now be studied by examining the intersections of the cylinder and torus.
The RCGR Mechanism and Its Surfaces
We turn now to consider what is the best orientation of a triad of coordinate axes, xyz, to which to relate the mechanism. First the general configuration of the mechanism must be defined, and all parame- 18
ters must be in their most general form. Figure 3 shows the general
configuration of an RCGR mechanism. Angle X is the angle between the
(C) _ axis and the (R). „ axis, and B B is the common perpendicular cLj,n Xc o between them, The common perpendicular between the (R)lp axis and the
(R), axis is O'C, and angle ? is the angle between them. For conven ience, the common perpendiculars were incorporated as part of the linkages. Link #2, the reference link, is now link 0!B BD; link #1 is link O'CA j link #3 is link DA; and link ifk is A A. There are eight o o defining parameters; the six lengths O'B , B B, O'C, CA , A A, and AD; and the two angles \ and 5. The angle of rotation of link DA about the
(c)p axis is labeled angle g and is measured from the position of DA when it is parallel to B B. The angle of rotation of link A A about the (R)]_,-, axis (angle 0) is measured from the position of A A when it is perpendicular to the (R),? axis, as shown in Figure 3. Angle ¥', is the angle of rotation of link #1 about (R)-.p and is measured from the position of O'C when it is parallel to B B.
With these considerations in mind, the simplest configuration for the general orientation of the related torus and cylinder is shown in Figure k. As was suggested by Jenkins (3), the coordinate axes are positioned so that the equation for the more complex surface, the torus in this case, can be written in its simplest form. The z axis is therefore coincident with the (R)1P axis and passes through the center of the torus at 0. The xy plane is perpendicular to the z axis and contains line 0A . In general, the cylinder axis is located randomly in this (x,y,z) space; bat whatever its position, a common perpendicular
(B B for the linkage illustrated in Figure 3) can be drawn between it 19
z axi
(R) 1:2
Figure 3. The RCGR Mechanism Figure k. General Orientation of Torus and Cylinder 21
and the z axis. Since the torus is a surface of revolution, the x and y axes may be rotated about the s axis to any desired position; and it is convenient to make the rotation so that the x axis is parallel to B B. o The relations between the torus-cylinder parameters and the mechanism parameters are as follows; r = A A* X = Xq s = DA- q =BB, o ' o
The remaining correlations are not as obvious5 bat referring: to Figures
35 h$ and 5« The following relationships can be derived0
h = B 0' - CA cos 5 . o o
p a (CTC^ + A~C2 sin2 5) where p is the primary radius of the torus*
a » ? ~90£! where a is the angle of rotation of the plane of the torus generating circle abcut p? away from the vertical plane„ Angle a is illustrated in Figure 5»
Y = cos^CO'C/p) where y is tne angle of rotation cf the plane of the torus generating circle about a line through its center parallel to the z axis. Angle
V is illustrated in Figure 5° (When both y and a are equal to zero, the torus will be a right ci.rcui.ar torus with circular cross-section, so the two angles can be regarded as measures of the deviation away from that special case). The eight parameters of the torus™cylinder orienta- 22
z axis o a + 90
C
Figure 5. Aid for Defining Mechanism - Surface Relations 23
tion — r, 3l, s? q, h, p, CL9 and y — are now defined in terms of the mechanism parameters. Angle ¥ is the angle of rotation of p about the z axis and is equal to Y'-y.
Equations of Torus and Cylinder
Now that the relationships between the RCGR mechanism dimensions and its corresponding surfaces have been defined3 the equation for the torus and the cylinder will be derived in the xyz coordinate system oriented as shown in Figure 8, The equation for the cylinder is derived by picking a new coordinate system x'y'z'., such that the z ° axis is coincident with, the axis of the cylinder,, the x" axis is coincident with B B and positive in the plus x direction, and,, of course9 y" is perpendicular to the other two at their point of intersection.. The 2 2 2 equation of the cylinder can easily be written as x' "* y? = s in this coordinate system^ and then translated and rotated by standard techniques to the xyz coordinate system (see appendix). The equation for the cyli.nder is found to be
2 2 2 (x - q) + [y cos X + (z - h ) sin X] = s
The equation of the torus can be found by rotating a general point on the generating circle around the z axis; the result is^
/ 2 2 2 2 2 >2 (x + y + z - p - r ~ 2pz sin y tan a)
i 2 2 f 2 2 . 2 v = 4p cos y (;r - z sin DC J
The derivation of this equation is given in the appendix, CHAPTER IV
TORUS-CYLINDER INTERSECTIONS
Survey of .Intersections
The intersections of a cylinder and a torus can have a great variety of forms and properties,, and this means that an RCGR mechanism is capable of many varied motions. (That Is one reason why the RCGR combination was chosen for investigation). Perhaps the hardest task in discussing the intersection is organizing the various kinds of intersections into meaningful categories and relating this organization to the motion of the mechanism.
The intersection of a cylinder and a torus can have one, two5 three, or four closed loops. These loops may be separate or may have a common point between them, as there is in a figure eight* Tne number of separate loops represents the number of ways In which a given mechan ism can be assembled to form a linkage,, Common points occur in the limiting case as two separate loops approach being tangent to each other. The linkage will have a "point of uncertainty" at these common points^ or branch points., since there will, be three possible paths that it can follow.
In the following discussion* the angle of rotation ( T in Figure k) of the primary radius of the torus will be considered the input angle, and the angle of rotation of the cylindric joint (j3 in Figure k) will be considered the output angle. Generally, if the intersection i.s a single closed loop (Figure 6-aJ, it can encircle neither the torus nor the cylinder, The mechanism, then, must be a do able -rocker, since v\> and
3 must have rotations of less than 360 degrees,, The only exception is the degenerate case of two loops becoming coincide.!-* ana forming a circle, which happens only when the cylinder is tangent to the torus around their entire circumference. One case is shown in Figure 6-c, where the tan gent circle is along the inner wall of the torus; the other case occurs when the tangent circle is along the outer wall of the torus. The mechanism, in this special case, will be a double-crank. Figure 6-b shows that both input and output oscillations may be made to approach
360 degrees.
When the intersection is made up of 'two closed loops, the link age may be assembled in two different ways. The intersection may represent any one of the four following possible combinations, l) Both loops represent double-rocker mechanisms (Figure 7-a)« 2) Both loops represent rocker-crank mechanisms (Figure 7-b). 3) Both loops re present crank-rocker mechanisms (Figure 7~c)„ k) Both loops i-epresent double-crank mechanisms (Figure 7--d). If in case #1 (Figure 7-a) the cylinder is made tangent to the inner wall of the torus$ there will be one branch point. If the cylinder is also made tangent to the other side of the inner wall, there will "be two branch points. Similarly, by making one and two points of tangeney, cases #2 (Figure 7~k)» fe
(Figure 7-o), and #k (Figure 7-d) can also have one or two branch points.
When there are three closed loops5 the mechanism has three possible configurations. There can only be two combinations of three loops - - two loops represent rocker-crank mechanism and the other loop represents Figure 6-a. One-loop Intersection - Case a
Figure 6-b. One-loop Intersection - Case b Figure 6-c. One-loop Intersection - Case c I 3
Figure 7-a. Two-loop Intersection - Case a
Figure 7-b. Two-loop Intersection - Case b :••:
s / _
f I
Figure 7-c. Two-loop Intersection - Case c
Figure 7-d. Two-loop Intersection - Case d 30
a double-rocker (Figure 8-a), or all three loops represent double- rockers (Figure 8-b). The first combination can have one, twt>9 or three branch points. There can be one such point between the rocker- crank loopEor between one of the rocker-crank loops and the double- rocker loop (Figure 8-c,i. In the case of two branch, or double, points3 both can be between the rocker-crank loop? or one can be between the rocker-crank loops and one between the "inside" rocker-crank, loop and the double-rocker loop, (Note that when both double points are between the two rocker-crank loops, four closed loops will be formed*,
However, since it is actually the case of two closed loops becoming tangent to each other at two points9 the curve is still considred to be two closed loops)* When there are three double points? one is between the "inside" rocker-crank loop and the double-rocker loop,, and two are between the two rocker-crank loops. The second combination (all three loops represent double-rockers) can have only one double point, as shown in Figure 8-d, when the cylinder becomes tangent to the inside wall.
Four closed loops represent either four rocker-crank mechanisms
(Figure 9-a.) or four double-rocker mechanisms (Figure 9-b). The second combination can have one or two double points5 depending on whether the cylinder is tangent to one or both sides of the inside wall of the torus. The first combination can have from one to six double points.
The case of six double points is shown in Figure 9~CJ where the cylinder is tangent to the torus at six separate locations. The other cases may be visualized by changing the diameter of and moving the cylinder so that the number of points of tangency reduce by one until there is Figure 8-a. Three-loop Intersection - Case a
Figure 8-b. Three-loop Intersection - Case b Figure 8-c. Three-loop Intersection - Case c
Figure 8-d. Three-loop Intersection - Case d 33
w/-
Figure 9-a. Four-loop Intersection - Case a
Figure 9-b. Four-loop Intersection - Case b 34
V
\ S
Figure 9-c. Four-loop Intersection - Case c
Figure 10. Two-loop Intersection 35
only one point of tangency, It is believed that the preceding survey includes all possible
types of intersections. Every type,, however * may have several different-
forms. For example, the case of two closed loops,, both being double-
rockers, shown in Figure 7-SL9 can also have the form shown in Figure 10. (These two forms have very different properties. In Figure 10, the
intersection encircles 3^0 degrees of the torus generating circle and
angle Y has little displacement throughout each loop; while in Figure
11-a, the angular displacement of the generating circle (angle 0) may
be very small and angle f may be made to vary greatly in each loop*
If these properties had been of more interest than those designated
input and output, then the organization of types of intersections
could be changed to emphasize their variations). The presence and
location of double points was especially emphasized because they indi
cate the boundaries between types of intersections. In Figure 8-c9 if the cylinder is moved slightly to the right, the intersection becomes
two closed loops; and if the cylinder is moved slightly to the left,
the intersection Is three separate closed loops. It should also be
noted that in Figures 6-c, 8-b, 8-d, 9~h9 and 10, the radius of the cylinder must be greater than the radius of the torus generating circle,
When the restriction is not satisfied, the intersections shown in these
figures become one of the other given types and do not represent a
different case,
Dimensional Restrictions
Projections of the two surfaces can be examined to determine y
conditions of the surfaces' dimensions that must be satisfied if cer« tain intersections are to occur- (i.e. a three-dimensional Grashof
Criteria). Figure 11-a shows the projection of the cylinder and the torus in the xy plane, where
y •—• • 2 2 r = \J p -2rpcosY + r
2 2 r = \/ p + 2 r p cos y + r
These relations are derived from Figure 11-b by applying the cosine rule to triangles OA A and OA A 3 „ Note that when y » 0, r. = p - r and r = p •* r. As can be seen from Figure u, if q - s > r0, there d d. can be no intersection, and the linkage cannot be assembled. If q - s < r?, the occurrence of an intersection is possible, but not certain. If there is an intersection the following statements may be made, l) If q + s > r0, the intersection cannot encircle the C- cylinder, so the output must be a rocker,, 2) If q + s < r , the intersection may encircle the cylinder,, but does not necessarily do so,
3) If q - s > -r_ or if q + s < r , the intersection cannot encircle the torus, and the input cannot be a crank, k) If neither of the conditions in statement #3 are true^ then the input may be,, but is not necessarily a crank. The projections of the surfaces in the yz plane would yield additional conditions, but the critical distances become so complex that the advantage of simplicity is lost, Figure 11-a. Projection of Torus and Cylinder in xy Plane
OA » = T. OA = r, OA = p o A A» = r o A
Figure 11-b, Illustration Defining r and r, 38
CHAPTER V
INPUT-OUTPUT RELATIONS
Intersection Equation and Solution
The equations for the torus and the cylinder were derived in xyz coordinates at the end cf Chapter III. The equations of their inter section will now be used to find input-output relations. Both equations can be written in the form f (x, y, z) = 0. The equation of the torus will be symbolized by T(x:, y? z) = 0, and the equation of the cylinder by
C(x, ys z) » 0. The intersection of the two is given bys T(x, y, z) -C(x, y, z) = 0. In order to get the equation of the intersection of many surfaces into a useful form, both original equations can be written in the form z = f (x, y); and then these can be subtracted to give the intersection in the form l(x, y) = 0. This is the equation of the pro jection of the intersection in the xy plane. In the case of the equation for the intersection of a. torus and cylinder, this method becomes too complicated, since the resulting equation l(x3 y) =0 is an eighth, order equation which, cannot be solved in a closed form to give y as a function oi x, Iterative methods could be used to find y for stepwise small
Increments of x, and then z would be found from the forms z = f(x9 y). However, this would be a very lengthy or, in cases where all. iterative procedures diverge, an impossible method of solution,
The equations of the torus and cylinder can be expressed in terms of the following parameters: ?, the angle of rotation of the primary 39
radius of the torus; 0, the angle of rotation of the secondary radius
of the torus; 3, the angle of rotation of the cylinder radius; and d, the
length BD in Figure 3, page 19• (The length d is taken to be positive
above the common perpendicular between the torus and cylinder axes and
angle X can then be considered to vary only between »90° and +90°), The
advantages of this parametric form are that it reduces the order of the
final equation to be solved, and the equations are expressed in terms of
the input-output parameters: Y, 0, d, and 0. From the projection of the
torus generator in the xy plane, shown in Figure 12, the parametric equa
tion of the torus can be shown to be:
+ x = p cos Y + r cos 0 cos (Y + y) + r sin 0 sin a sin (Y Y)
y = p sin f + r cos 0 sin (Y + y) « r sin 0 sin a cos (T + Y) z = r sin 0 cos a,
The parametric equation of the cylinder, which can be found from geome
tric considerations, is:
x = s cos 3 + q y = s sin 3 cos X - d sin X
z = s sin 3 sin X + d cos X + h.
Along the intersection, the x, y, and z coordinates of one of the surfaces must equal, respectively, the x, y, and z coordinates of the other surface,
This results in the following three simultaneous equations:
s cos p + q = W
p cos T + r cos 0cos(Y + V) + r sin 0 sin a sin(T + y) 40
y axis
"U r sin <$> sin a
V ¥ .+ Y
x axis
Figure 12. Projection of Torus Generator in xy Plane 1+1
s sin |3 cos X - d sin X = (2)
p sin ¥ + r cos 0 sin (Y + y) - r sin 0 sin a COS(Y +Y)
s sin p sin X+dcosX+h=r sin 0 cos a. (3)
These three equations ccntain the four unknowns : Y, p, 0, and d,
By letting the input angle, Y, vary from 0° to 3^0°, the other three
unknowns can be found from these equations for each value of Y, thus
defining the input-output relations. To obtain d, equation (3) is written in the form;
d - (l/cos X)(r sin 0 cos a - h - s sin 8 sin X) (h)
Substituting this into Equation (2), an expression for |3 is found;
-1 sin 3 = —Fsin I (r sin 0 cos a - h) + p cos X sin Y (5) s
+ r cos 0 sin (Y + V) cos X - r sin 0 sin a cos(Y + V) cos X]
Finally, from equatiors (l) and (5) the result is a fourth order equation in cos 0.
h ? 2 tr\ cos 0 + A cos 0 + A cos 0 + A cos 0 + A. = 0 (oj where
A - 2(AB + CO) 1 ~ 2 2 ' -1 A + G te
2 2 2 B + 2AC - G + E 2 2 2 A + G 2(BC - GE) A = N- - 5 A2 , „2 A + G 2 2 A, C -E k 2 2 A^ + G
where
B = 2(1^ M + Mg M^)
C = M^ + % + M^ + M^ - s<
E = 2 (ML M^ + Mg M )
G = 2(1^ ^ + \ M ) where
ML = r sin X cos a - r sin a cos(f + Y) cos X
M = r cos X sin(Y + A0
M = p cos X sin Y - h sin }. 5
M^ = p cos(Y + V)
M_ = r sin a sin(Y + V)
M/ = p cos Y - 1. ^3
The quartic equation is here solved by closed, or classical methods, rather than approximate or iterative methods, even though some consider the latter methods easier to use. There are no such approximate methods that can be used for all quartics with any combinations of coef ficients; and, in many cases, the iterative procedures diverge or become very long and would use excessive computer time (6). On the other hand, there are several classical methods for solving a quartic9 or fourth order3 equation. The one used in this paper was discovered by Ferrari (see reference 8, pp 82-98) • Briefly., this involves transposing terms of the quartic so that the two terms of fourth and third order are set equal to the remaining terms. By the addition of an artificial term containing an unknown, both sides of the transposed equations can be made perfect squares; thus the squareroot of both sides can be taken,, reducing the quartic equation to two quadratics. The unknown in the artificial term is found from the "resolvent" cubic; that results from the conditions that the artificial term must satisfy.
Another solution used is one developed by Neumark (6). In this3 the quartic is artificially factored to two quadratics. The relations that must exist between the coefficients of the quadratics and those of the quartic are used to define an unknown that can be solved by a
"resolvent" cubic equation that results. The coefficients of the two quadratics can then be found in terms of the coefficients of the quartic«
Once the value of cos 0 has been found with the aid of a digital computer, the value of sin 0 must be examined to see whether it is positive or negative, before the value of the angle 0 itself can be determined,,
The quartic equation in cos 0 resulted from an equation in which cos 0 kk
and sin 0 were the only unknowns, for a given value of Y. The value of sin 0 can be determined from this equation, and, thus, the value of 0 can be found. Once 0 is found, cos p is obtainable from Equation (l) 9 and sin 3 is obtainable from Equation (5). The value of d may now be found from Equation (h). However, if the constant angle X is close
"to t 9°° J then d will be indeterminate in this equation, and Equation (a) must be used to find d. (it must be remembered that these values of
Y, 0, d, and $ form an associated quadruple, and such a quadruple of values may exist for every Y Ln the interval 0 < Y < 3600).
A computer program was written to perform the preceding operations with angle Y being increased stepwise from 0° to 3600 (see appendix),
First, any values of the eight parameters that define orientations of the cylinder and torus (also defining the eight parameters of the RCGR mechanism) can be used a;3 the input data, and then the three outputs- angle (B, angle 0, and length d are calculated in terms of Y. These values, then, could be used to find the x, y, and z coordinates of the curve of intersection, if this were desirable. But this is not done here .
Illustrated Results
The program was executed with several sets of orientation parameters in order to illsutrate the different types of intersections discussed in Chapter IV. The fact that the parameters can be chosen to illustrate specific examples of intersections is evidence of the use fulness of the method of generated surfaces for synthesis. For example, it was desired to obtain an intersection similar to that shown in Figure 7-c? page 29, Using intuition and simple geometric considerations, the
following data were chosen for case #1: s = U.O, q = 0.0, X = 1.6
radians, h = 3.5? r - 1-0,p = 2.0, Y = 0.0, a = 0.0 The results are
shown in Figures 13-b and 13-c . Different loops are represented by
different kinds of lines, broken or solid. Note that for a double
point to occur, the values of 3 and d for one loop must equal, respec
tively, the values of 3 and d for the other loop at a single value of
Y. Thus, in Figure 13-h, although the curves representing 3 and d for
the two loops cross each other, there is no double point since they
cross at different values of Y; and the two loops are completely separate,
Even though angle 0 was not considered to be an output, the program
can calculate its values, and it is plotted in Figure 13-a.
Figures ik-a, and ik-lo show the output plots of case #2, ar.
intersection similar to that shown in Figure 8-c, page 32. There are
three closed loops and a double point at Y - 0° (or Y = 3^0°). Loops #1
and #2 almost meet to form a double point, and loop #3 is very close to
becoming two loops joined by a double point, The values of the parameters
for this case are: s = 0.5, q = 1.0, X - 1.570? radians, h = 1.5, r = 2,0,
p = 3.0, Y = 0.0, a = 0.0.
Figure 7-d, page 29, can be exemplified by the da~;a; s = 2,0,
q - 0.1, X = 0.05 radians, h = 0-3, r = 1.0, p = 2.0, Y = 0.05 radians, a = O.05 radians; and the results fpr this case #3 are shown in Figures
15-a, 15-h, and 15-c. This case shows the effects of angles y and a,
If they had both been equal to zero, it could have been predicted that
: 0 3 and 39 would cross at approximately Y = 90° and Y = 270 (as will be 220-1
200 4- i+oJ 0 (deg) 20-^ Figure 13-a. Angle 0 versus Angle Y - Case #1 240 -i 9° *<**) l8° 27° 36U Figure 13-b. Angle p versus Angle Y - Case #1 -3-0 J Figure 13-c. Length d versus Angle Y - Case #1 320 3^0 360 Y(deg) Figure 1^-a. Angle g versus Angle Y - Case #2 U9 ^1 -1-1 +1 (units) -2 - +2 -3-1 Jj -h - +h -5- -^ 0 k0 Y'deg) 8o f- / '28h '.$)0 320 360 Figure lif-b. Length d versus Angle Y - Case #2 160 120 (deg) 80 . \ 2 40 - -/ £'28 c 320 360 1+0 y(deg)80 Figure l4-c. Angle 0 versus Angle Y - Case #2 5 0 100 90 - 80 - ^(deg! (cleg) 90 180 270 360 -80 • -90 . -100 Figure 15-a. Angle $ versus Angle ^ - Case #3 .8o n •70 A d (units) .60 J 90 wfA y 180 270 360 f(degj -1.20- -1.30 -1.40 -J Fig-are 15-b * Length d versus Angle Y - Case #3 52 (deg) ibo 360 Figure 15-c. Angle 0 versus Angle Y - Case #3 seen in the next case). However, with the given values of V and a, the result is that shown in Figure 15-b. Figure l£ describes a case $k in which angle X has been increased to make 3 and p more distinct, and angles Y and a are both zero* The plots of 3 and 3p cross at approzimately Y = 90° and f = 270° as expected. The data for Figure 16 are; s - 2,0, q - 0.3, X = 0.5 radians, h « 0.3, r - 1.0, p = 2,0, Y « 0.0, a = 0.0. The intersection which has six double points (Figure 9-°* page 34) is, of course, a very special case. The radius, s, of the cylinder must be equal to r and also to p - r. Angle X must be 90° j and h, q, y, and a must equal 0. The values s = 1, r - 1, and p = 2 are chosen and the result is case #5j shown in Figure 1'7. The coordinates of the six double points are; f = 0 (360), p = 0 (360) ; Y = 90, 3 = 90; f a 90, 3 = 270; f * 180, £ = l8C•; y = 270, 3 = 90; Y = 270, 3 = 270. The "loops" representing d through d< do not appear to be loops because their other sides are directly under the visible lines (at different 3 values). When s - 4.0, q - 0.5, X = 1.4 radians, h = 0.3 r = 1.0, p = 4.0, y = 0.0, and a •-• 0.0., bhe output is shown in Figures 38-a and 18-b (case #6) . There are two closed loops, and this case is similar to that shown in Figure 10, page 34, with the intersections on opposite sides of the cylinder. This survey exhibits just a few of the possible outputs from RCGR mechanisms. However, the computer program shown in the appendix can be used to compute the output from any combination of links that form an RCGR mechanism. If the input angle ¥ is assumed to rotate at a 2 -, Figure l6-a. Length d versus Angle ¥ - Case #h 360 -) 270 (deg) 180 - 90 A 90 l*> ,(deg) 270 Figure l6-b. Angle 0 versus Angle ¥ - Case #4 3 units) 0 180 270 360 -2. >< d \ « Y(deg) -4 Figure 17-a. Length d versus Angle Y - Case #5 (deg) 90 180 Y(deg) 270 360 Figure 17-"b. Angle p versus Angle Y - Case #5 210 195 180- 165 3 (deg) 15 / 1 Y 1 JT 1+0 \ko 180 !20 360 Y(deg) -15 • V -30 Figure l8-a. Angle 0 versus Angle T - Case #6 v,- 5> 2- d (units) "iV +J- -/f J y kO ' 120 200 240-Ih 32 0 ^60 Y(deg) \ di -2 ^ -if J Figure l8-"b. Length d versus Angle Y - Case #6 :il 58 constant velocity, all of the preceding graphs may also be used to estimate velocities by analogy to the slopes of the curves. Thus, quick-return effects and dwell periods of these mechanisms can be examined. CHAPTER VI CONCLUSIONS The characteristic motions of an RCGR mechanism can be represented by the intersections of a cylinder and a torus, having any sizes and relative positions. A given RCGR linkage may be assembled in from one to four different ways and may have from one to six "double" or "branch" points, the numbers depending upon the specific dimensions of the com ponent links. It can be determined by inspection of the intersection whether the mechanism is a double-crank, double rocker, rocker-crank, or crank-rocker; and certain dimensional limitations for these cases can also be established. With the aid of a digital computer, a parametric form of the equation of the intersection can be used to calculate stepwise small displacements of any representative linkage. Although the method of generated surfaces is not as widely applicable as some other methods, it can be a great aid in the investiga tion of certain mechanisms. The simplicity and directness of this approach make it readily available to designers and investigators, and it is directly related to the physical aspects of the mechanism being studied and its motion. APPENDIX A DERIVATION OF TORUS AND CYLINDER EQUATIONS Derivation of the Torus Equation As stated in Chapter III, the equation of the torus is derived by rotating the generating circle about the z axis. The center of the generating circle is located a distance p from the origin and, for convenience, is placed aLong the y axis to derive its equation. It has been shown that, in the general case, the plane of rotation of r, the generating-circle radius, is inclined away from the yz plane by an angle Y about a line through its center parallel to the z axis and by an angle a about the y axis. The equation of the plane is found from its x, z, and y intercepts to be: £_ + Z z_tan_a __ , , p tan Y p p sin Y ' x K J The generating circle is defined by the intersection of this plane with a sphere whose radius is r and whose center coincides with the point y = p. The equation of the sphere is x2 + (y - of + z2 = r2 (2) Solving Equation (l) for x yields x = -(y - p) tan y + z tan a sec y (3) 6l and substituting this expression for x into the Equation (2) gives y as a function of z on the generating circle, as shown below. 2 2 o o o p o (y-p) sec y - 2 z(y-p) tan Y sec y tan a + z tan a sec Y + z - r = 0 which can be regarded as a quadratic in (3/ - p), and simplified to the following forms o (y-p) - 2z(y - p) sin y tan a (4) 2 2 2 2 2 + z tan a^cos Y(Z - r ) = 0 Solving this for y yields: ! p p p '?'?'?'? y = z sin V tan a + p + vz tan a (sin y - l) - z cos Y + r cos y or / 2 2 ? • v y = z sin y tan a + o + cos Y vr - z sec a (5) Any arbitrary point on the generating circle will be some distance, £, from the origin; and as the generating circle is rotated, the equation 2 2 2 t = x + y must be satisfied. Solving the Equation (2) of the sphere for (x + y ) yields; x~ + y = r - z + Spy - p - or ^ - r2 - z2 + 2py - p2 (6) for an arbitrary point on the sphere. To make this point on the sphere also a point on the generating circle, the expression (5) for y in terms of z on the generating circle is substituted. 2 2 2 2 f~2 2 2 2 I = r - z~ + 2 pz sin Y tan a + 2 p + 2p cos y \Jv - z sec a -p and thus 2,2 2 2 2 x +y + z - r - p /2 2 2 2 p z sin Y tan a + 2 p cos Y Vr - z sin a (7) is the equation of the torus. Derivation of the Cylinder Equation The equation for the cylinder in the xr yf z' coordinate system oriented as described in Chapter III is simply 2 2 2 x ' * y' « s (8) The transformation from the x' y' z' coordinate system to the xyz coordinate system of Figure 8 is most easily made in two steps: (l) rotating the x' y' z' system counter-clockwise about the x* axis through an angle X so that z' is parallel to z and (2) translating the rotated system (say x" y" z") a distance q along the x' axis and a distance h along the z axis so that it coincides with the xyz system. The relationships between the xyz coordinates and the x! y3 z1 coordi» nate s can be found as follows * x' - x'r y1 = y" cos X + z" sin X z' = z" cos X - y" sin X x" = x - q (10) x it = y z" = z - h Combining the se: x1 = x - q (11) y' =y cos X + (z; - h) sin X z1 = (z - h) cos X - y sin X The equation of the cylinder, then, is (x - q)2 + [y cos \ + (as - h) sin xf = s2 (12) APPENDIX B COMPUTER PROGRAM General Comment The input data that must be supplied to the program are s(S) q(Q), \(IA), h(H), r(R), p(BK>, v(GA)3 a(AL). The terms in parentheses indicate the way the data are labeled in the program. (For card inputs the data do not have to fit any format,, but they must be punched in the given order with a comma following every value). The values of the angles in the input data (\, Y, and a) mast be in radians, The card labeled AGAIN" determines the increment in angle Y, labeled PSI in the program. In the program shown, the increment is rr/l8o radians or 1° . For the output, the variable angles are converted to degrees and the length.s are in the same units as lengths used in the input. 65 1 ( MP DATA IN - S, Q, LA, H, ~R? RH, GA, "AIT , PSI = 0 COMPUTE Ml, MS, M3, M^3 M5, M6 COMPUTE A, B, C, E, G i NO YES 2 SOLVE RESULTING QUADRATIC FOR REAL ROOTS - PH[l]9FH[2] COMPUTE QUARTIC COEFFICIENTS - AX, A25 A3, Ak COMPUTE REDUCED CUBIC COEFFICIENTS -BI, B2, B3 SOLVE CUBIC EQUATION FOR REAL ROOT - CAT J_ USING CAT, SOLVE QUARTIC EQUATION FOR REAL ROOTS PH[l]: Hi [2], Hi[3 ], EH ft] - THE FOUR VALUES OF COS (f> I COMPUTE SIN 0 (SPH) USING THE EQUATION THAT WAS SQUARED TO OBTAIN THE QUARTIC [COMPUTE THE ANGLES $ (ANGPHTII) COMPUTE COS 3 (CBET) USING EQ. (l) PG. 39 COMPUTE SIN B (SBET) USING EQ. (5) PG. kl COMPUTE THE ANGLE g (BET[l]) NO I ^ - 1.0 > o? YES COMPUTE d (LE) USING EQ. ft) PG. 4x| [_COMPUTE d (LE) USING EQ. (2) PG, kl (WRITE _PSIa PHI, BETA, IE PSI = PSI + n/l80 1 NO Logic Flow Chairt LABEL OOOOOOOOOPRINT 00 1 68 1 427C0MP RE XXXXXXX/HPGMESU ALGOL BE GIN FILE I N HPGIN (2*10)* FILE 0 UT HPGOU T 6(2*15)2 RtAL A RRAY PHPA NGPH^LEPBETC0JA3 I INTEGE R I 5 REAL SPQPL A*D*H>RpRH»6A»AL»PSIrCPH#SPHpCHt*A*B#C*GpEpGl*Al#A2#A A4,8i *B2pB3*CUc2*CTHET,CnTH,THET,THETA0,ANGt#ANG2,ANG3#PA CAl^C A2i>CA3>CAT>KP*DA,EA;>XXpMl,M2j,M3>MA,M5#M6pCC;,DDpAApBB> CBET* SBETJICHECPPIEJ>VERT*CHE1PATE*DEGPOA1| LABEL ALLPR EPPTR Y#AGAIN»G0aDApG00DB,G00DC*ANSWAftlS2*ANS3,ANS4*0K NOWPF INEPKI EEP*MORE^HELP*SPEJ- LIST 0 ATAIN CS> QPLA, H,R,RH»GApAL> ' 5 W W FURMAT OUT FOTi ("DATA IS- - S = PF70?PX4P Q = "PF7<,2PX4P"LA = *%F794PX4P"H F703, X4P"R=«PF7,3*X4P"RK="pF7o3pX4*"GA=«pF7e5pX4*"AL="p F7e5/ /)! FURMAT OUT FOT? CX6p* P$1"P X6P"PHI1/3"I>X4, "BETA 1/3% X5P "LEI/3"* X6p "PHI? /4%X 5p"BETA2/4"pX4p"LE2/4"/) ) FORMAT OUT F0T4 C" SOMETHING IS WRONG") FORMAT OUT F0T5 ("THE CYCLE 15 COMPLETE"//) i FORMAT OUT F0T6 »AJK 6(.4#X3»F6c4#X3pF8o4pX3pF8eftpX3pF8a4*X3*F9«4#X3* F . .4 ) ? FORMAT DUT F0T7 t*l$, F8o4vX3pF64iqj,X3pF8,4Jtx3pF8o4pX3pF804pX3pF8o4/> I FURMAT OUT F0T8 CX2PE 9S3*X?PE983PX2PE9,3PX2*E9,3PX2PE993) * HELPt READC HPGFN P/,OATAIN)[MORE] ; HKITEC HPGQUTtNO 3)1 WRITE (HPGOUTPF DT1*D ATAIN) I WHITE (HPGOUTPP OT25 I P SI • 0 } REP: D*PSI + GAJ PIE + 3 ,1415 926536 J Ml*Rx SIN(L A)xCOsCAL5-RxSINCAL)xcOS(0)xCOS(LA) M2«-Rx COSCL A)xSlNCO) J M34-RH XCOSC LA5XSIN(PSI)-HXSIN(LA) J M4«-Rx COSCD ) J M5«-Rx SINCA L)x5lN -q H 1* Denavit, J„, and R. S„ Hartenberg, Kinematic Synthesis of Linkages, McGraw-Hill Book Company, Inc,,, New York, 196^, PP» 31-38, and 3^3-368. 2. Hunt, K, H., and J, Ra PhilJips, "On the Theorem of Three Axes in the Spatial Motion of Three Bodies," Australian Jl« of Appl. Science, Vol. 15, (196*0 pp. 267-287. 3. Jenkins, E. M., Ends of Travel, Locking Positions, and "Uncertain Motion" in Spatial Mechanisms, Thesis, Georgia Institute of Technology, 1967* he Johnson, H. L., An Approximate Synthesis of the Spatial Four-bar Linkage, Ph.D, Dissertation, Georgia Institute of Technology, 1964/ 5„ Meyer zur Capellen, W., "Kinematics - A Survey in Retrospect and Prospect," Journal of Mechanisms, Vol, 1, Wo. 3 and 4,(1966). 6. Neumark, E., Solutions of Cubic and Quartic Equations, Pergamon Press, Ltd., London, 1965* pp. 5-l4. 7. Salmon, G„, Analytic Geometry of Three Dimensions, 7th ed,, Vol, I, Chelsea Publishing Co,, New York, 1927, pp. 1-39- 8. Uspensky, J. V., Theory of Equations, McGraw-Hill Book Company, Inc., New York, 19^" pp. 82-98. 9. Yang, A.T., Application of Quaternions and Dual Numbers to the Analysis of Spatial Mechanisms, Ph.D. Dissertation* Columbia University, 1963fl