DESLGN CALCULATION FOR FOUR BAR
UNKAGE FUNCTEON GENERATORS

Thesis for fine Degree 05 Din. D.
MICEEGAN STATE UNIVERSE?Y
Jerome Chmieiewski

1961

THESIS

This is to certify that the

thesis entitled
DESIGN CALCULATION FOR FOUR BAR LINKAGE

FUNCTION GENERATORS

presented by

Jerome Chmielewski

has been accepted towards fulfillment
of the requirements for

Ldegree inL

07.2.1.1? 77 )1... Ac;

Major professor

 

’1) Law-‘2 2.4%}

Date

 

0169

   
     

  

LIBRARY

Michigan State
University

 

ABSTRACT

DESIGN CALCULATION FOR FOUR BAR LINKAGE

FUNCTION GENERATORS

by Jerome Chmielew ski

Since a four bar linkage has only one degree of freedom, the
output displacement angle of the linkage is an intrinsic function of the
input displacement angle. Because of this fundamental property of
four bar linkages, it is possible to design a four bar linkage to have its
two displacement angles (input and output) satisfy an arbitrary functional
relation at up to and including five precision points. That is, the in-
trinsic function can be made to approximate an arbitrary function over
a given range of precision points, and the intrinsic functional relation
and the arbitrary functional relation will each be satisfied by the pairs
of displacement angles at the several precision points. Thus, the four
bar linkage may be designed as a function generator of an arbitrary
function.

In the same vein, it is possible to design a four bar linkage
function generator that has precision derivatives at one or two of its
precision points.

In the design of a four bar linkage function generator that has
five precision points, the ratios of each crank link and the coupler

link to the separation link, the starting position of the linkage, and the

Jerome Chmielewski

five pairs of displacement angles which are the precision points of the
generator must all be compatible with one another. The starting posi-
tion of the linkage is given by a pair of starting angles. These starting
angles must be determined in each generator design and they depend
upon the pairs of displacement angles specified as the precision points.

After the starting angles have been determined for a function
generator, the link ratios may be determined. The determination of
the starting angles and link ratios completes the design of the function
generator. That is, the starting angles and the link ratios completely
define a function generator up to its actual physical size.

The thesis presents a design calculation for four bar linkage
function generators that have five precision points. In the calculation
one of a pair of starting angles of the linkage is determined from the
solution of a cubic equation in its tangent. The remaining unknown
starting angle of the pair is then evaluated directly. Although this
design calculation is rather lengthy, it is straight forward and requires
no iterations or graphical solutions. The calculation is inherently
accurate.

The thesis also provides a design calculation for four bar linkage
function generators that have precision derivatives at precision points.
It is also implied that the design calculation can be modified to accept
higher order derivatives at each of two precision points.

Further, it is shown that there is a possibility in a given generator

Jerome Chmielewski

design that three pairs of starting angles may exist. When three
pairs of starting angles exist, they provide three different four bar
linkages which are capable of generating the same arbitrary function
with the same specified precision points.

Two problems, solved in detail, are included in the thesis.
The solutions of these problems demonstrate the calculation and its
accuracy. In the first solved problem it is determined that only one
four bar linkage generator exists for the five specified precision
points. In the second problem it is determined that three four bar
linkage generators exist that are capable of satisfying the five pairs

of displacement angles which are the specified precision points.

DESIGN CALCULATION FOR FOUR BAR LINKAGE

FUNC TION GENERATORS

BY.

Jerom e ’Chmielew ski

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOC TOR OF PHILOSOPHY

Mechanical Engineering Department

1961

Copyright by
Jerome Chmielew ski

1961

ii

ACKNOWLEDGEMENT

I wish to express my sincere appreciation to Dr. Rolland T.
Hinkle for his oft-given advice and encouragement which proved to be
invaluable in the development and in the final preparation of this
thesis.

It is also with pleasure that I here acknowledge my indebtedness
to the Consumers Power Company of Jackson. Michigan for the
financial assistance I received as a recipient of a Consumers Power
Company graduate scholarship.

I would like to thank Mrs. Allen Wonch for the excellent typing

of this thesis.

Jerome F. Chmielewski

TABLE OF CONTENTS

LIST OF TABLES ....................................
LIST OF FIGURES ......... . ....................... . . .
INTRODUCTION .....................................
CHAPTER I
ANALYSIS ..........................................
1. General Information
2. Notation
3. Intrinsic Equation

4. Determination of the Starting Angles

5. Determination of the Link Ratios

CHAPTER II

CALCULATIONS ....................................

1. Solved Problem 3

2. Problem Discussion

CHAPTER III

SPECIAL CASES ........ . . . . . . ......................

1. Precision Derivatives

2. Four Precision Point Design

iii

Page

18

23

23

49

53
53

56

iv

Page
CHAPTER IV
CONCLUSION .......................................... 58
1. Five Precision Point Design Calculation 58
2. Four Precision Point Design Calculation 59
3. Precision Derivatives 59
BIBLIOGRAPHY ........................................ 60
1. References Cited 60

2. Reference Reading 60

LIST OF TABLES

Table Page
I Expansion of ResultantA ...................... 19
II Coefficients of 8th Degree Polynomial ......... 21

III Precision Derivative Modification ............ 57

vi

LIST OF FIGURES

Figure Page
1. Four Bar Linkage ........................ 8
2. Four Bar Linkage of Example (1) .......... 45
3. Four Bar Linkage of Example (2), Case I . . . 46
4. Four Bar Linkage of Example (2), Case II. . . 47

5. Four Bar Linkage of Example (2), Case III . . 48

IN TRODUC TION

A four bar linkage, as its name implies, consists of four links
which are simply joined at four pinned joints to form a closed kine-
matic chain. When one of the links is held fast, the four bar linkage
is a mechanism, and two of four links are cranks, one is a coupler,
and the fixed link is a separation link. The two pinned joints at the
ends of the separation link are crank pivots. The separation link is
often merely a separating distance between two fixed pinned joints that
are the crank pivots and are attached to a stationary frame. Since an
end of each crank is attached to one of the separated crank pivots about
which the crank may rotate and the cranks are coupled at the opposite
ends at the pinned joints with the coupler, the mechanism is assembled
in a manner that permits only one degree of freedom. Motion is
generally imparted in the linkage by the rotation of one of the cranks
about its crank pivot, and input and output are generally given in terms
of the crank displacement angles which are measured from an initial
position of the linkage.

Because of their simple construction and operation and their
inherent amicability toward analysis, four bar linkages have achieved
widespread popularity in the field of kinematic design. The linkages
are used extensively in instruments and machines where the transfor-
mation of motion is a necessity or where a particular type of motion is
to be generated as an end in itself. Needless to say, there exist different

classes of four bar linkages.

One class of four bar linkages consists of those four bar linkages
that are designed for coupler action. A four bar linkage in this class
is designed to have a tracer point affixed to the coupler link execute a
prescribed motion when motion is imparted in the linkage. Typical
examples of these linkages are the Cyclodial Linkage and Robert's
Linkage. Each of these linkages is designed to have the tracer point
approximate rectilinear motion as one of the cranks is rotated about its
crank pivot.

Another class consists of four bar linkages that are designed for
coupler position. A linkage in this class is designed to have the coupler
link occupy a series of prescribed positions as motion is imparted in the
linkage.

The four bar linkages of yet another class are designed for the
purpose of generating particular functions in terms of the input and
output displacement angles of the linkage. It is to this type of four bar
linkage that the following discussion applies.

Since a four bar linkage has only one degree of freedom, the
output displacement angle of the linkage is an intrinsic function of the
input displacement angle. This intrinsic function depends entirely upon
the ratios of each crank link and the coupler link to the separation link.
The displacement angles have the character of variables when motion
is imparted in the linkage, thus form a discrete pair of values for each

position of the linkage.

When up to and including five discrete pairs of displacement angles
have been specified as precision points, one or more four bar linkages
can be designed to fit together in each of the various positions of the
linkage obtained by measuring the specified displacement angles from a
common initial position of the linkage. This initial position or configur-
ation depends upon the displacement angles specified as precision points,
and there is no guarantee that any of the four bar linkages will move
freely from one position to another. That is, there is no guarantee that
the intrinsic functional relation between real values of the input and output
displacement angles will be continuous for any of the four bar linkages.

Because of the property of four bar linkages stated in the above
paragraph, a four bar linkage can always be designed to have its crank
displacement angles satisfy an arbitrary functional relation at several
specified precision points, and if the designed four bar linkage has a
continuous intrinsic function, the intrinsic function will approximate
the arbitrary function. The intrinsic functional relation and the arbitrary
functional relation between the displacement angles will be satisfied
simultaneously at the several precision points. Thus, the four bar
linkage will generate the arbitrary function.

The design or synthesis of four bar linkages for use as function
generators has only rather recently been put in analytical form by

Freudenstein (1)" in athree, four, and five point approximation, and

*Numbers refer to references cited on page 60.

in an nth order approximation for n = 5, 6, and 7. In an nth order
approximation an arbitrary function and its first (n-l) derivatives are
equal to the intrinsic function and its first (n-l) derivatives at a single
precision point. Freudenstein (1) has shown that the crux of the five
precision point approximation is the determination of a pair of starting
angles which give the initial position of the synthesized four bar linkage
generator. Freudenstein (1) determines these starting angles by solving
two complicated higher-order simultaneous equations graphically or by
an iterative process with the aid of a digital computer.

It was felt that an investigation into the problem of designing four
bar linkages for use as function generators would prove justifiable if
the investigation provided a reasonably straight forward solution to
the five precision point problem, a procedure for designing function
generators with precision derivatives at more than one precision point,
and a determination of the number of different four bar linkages that

can be constructed upon the same five precision points.

CHAPTER I

ANALYSIS

1. General Information. The four bar linkage shown in Fig. l

 

page 8 can be designed to have its two variable displacement angles
satisfy an arbitrary functional relation f at five discrete precision

points . That is

$1 = f(¢i) for i =1, 2, 3, 4, and 5 (l)
where
f is an arbitrary function relating 4/1 to chi
1,111 is the angular displacement of link Ll from a starting position
as indicated by the starting angle ¢ measured from the x-axis,
¢i is the angular displacement of link L3 from a starting position

as indicated by the starting angle ¢ measured from the x-axis,

In order for equation (1) to hold true for each of the five precision
points, there must exist a certain compatibility between the four bar link
ratios Ll/L4’ LZ/L4, and L3/L4, the starting angles ¢ and g0, and the
displacement angles (Pi and “’1 for i = l, 2, 3, 4, and 5. Surprisingly, there
may exist more than one four bar linkage that will satisfy equation (I) for
the five precision points (4)1, 471). The determination of the starting posi-
tion of links L1 and L3 as indicated by t/J and (p rests upon the solution of

a cubic equation in wt or ¢t° Consequently, there will exist one or

three solutions, as given by the real roots of the cubic equation.

2. Notation. It will be convenient to use the following notation.

, = cos ,

¢1C (1)1
, = sin ,

¢1s ¢1
(P = tan 4)

t

. = cos #1.
1c 1

(/1, = sin (,0_
IS 1
wt = tan (,0

(ix-w.) = cos (4mm
1 1 C 1 1

«xx-w.) = sin hit-«0.)
l l 1 1

S

(«p-wt = tan (ch-v)

HM) + (<Pi-Willc = cos [up-v) + (4»i win

An expression of the type

- f ' 2' k d
Wis' wic' 1. (‘Pi Wilcl or 1 j, , m, an n

ori =j, k, m, andp
shall represent a determinant which has elements as indicated by the angle
symbol and indices. That is, the first index and Greek symbol shall indi-
cate the angle and the second index shall indicate a trigonometric function
of that angle for any element. It should be observed that in each deter-
minant each element in the third column is equal to unity. For instance

fori =j, k, m, andn

 

43.5. wk. 1. (4>J.-wj)c
(pks’ wkC’ 1' (¢k-wk)c
(pis' wic' 1’ (¢i"pi)cl : ¢ms some 1. (chm-«1%)C
¢ns, wnc' 1’ (¢n-wn)c
sin cpj, cos wj, 1, cos (¢j-¢j)

sin ¢k' cos wk, 1, COS “Pk-Wk)

sin 42m, cos Wm, 1, cos Wm-Wn)

 

sin on, cos Wu, 1, COS Wn-Wn)

The meaning of j, k. m, n, and p will be explained later in paragraph 4.

 

3. Intrinsic Equation. From Fig. (1) it is seen that
Z Z 2 2
L2 - (P1132) ~(X2-X1) + (Ya-Y1)
where
xl : LWWI-win
= +
Y1 Llw in)s
x2 = L4+L3(¢+¢i)c
= L
Y2 3(<!>+<1>].L)s
or

2 _ 7- 2
L2 — [L4+L3(¢+¢i)c - Llwwilcl + [L3(¢+¢i)5 ' Incl/Wig] '

 

 

P1(x )

1' y1

 

 

 

x
Fig. 1. Four Bar Linkage
After simplifying, the above equation may be put in the formi<
Q : -
ich + ll”1(sz + R3 ”)1 “We (7-)
where
: L 2
R1 4/L1 ( a)
: -L L 2b
R2 4/ 3 i l
R = [<L /L )2- (L /L )2+(L /L )2+ 1]/(L /L )(L /L) (2c)
3 l 4 Z 4 3 4 l 4 3 4
4’1 = (P + 431 (2d)
{Pi = W + £01 (28)

* This form is given by Freudenstein (1). If the values of q’ic and lilic
given in (2d) and (2e) are substituted in equation (2) and if the equation
that will result from this substitution is solved for l/Ji in terms ¢., then

. . . . . . 1
«pi W111 be defined as an intrinsm function of (Pi.

P1(x )

1' V1

 

 

 

x
Fig. 1. Four Bar Linkage
After simplifying, the above equation may be put in the form"<
Q : -
ich + 4&ch + R3 ((Pi LlJi)c (2)
where
R1 = L4/L1 (2a)
7- - 2
R2 1.4/1.3 ( b)
R = [<L /L )2- (L /L )Z+<L /L )2+ 11/(L /L )(L /L) (ac)
3 l 4 2 4 3 4 l 4 3 4
Q1 = (l) + (P1 (2d)
$1 = (I! + wi (26)

* This form is given by Freudenstein (1). If the values of éic and 411C
given in (2d) and (2e) are substituted in equation (2) and if the equation
that will result from this substitution is solved for (lg in terms 4)., then

. . . . . . 1
‘01 Will be defined as an intrinSic function of ¢i°

4. Determination of the Starting Angles. Now equation (2) is a linear

 

equation in R1. R2. and R3, and if i takes the values 1, 2, 3, 4, and 5,
there will exist a system of five equations in three unknowns which
may be regarded as consistent. With this in mind consider the set of
four equations produced by letting i take the values j, k, m, and n, and
also consider the set of four equations where i has the values j, k, m,
and p. The quantities j, k, m, n, and p are each to take a different
value from the set of five points 1. 2, 3, 4, and 5. If each set of four
equations is to be consistent, then the resultant or eliminant of each

set must vanish. Let the resultant of the first set be A, and the

resultant of the second set be A'. Then
A = |(¢+¢i)c. (¢+wi)c. 1. [(¢+¢i) - (w+wi)]c| = 0 (3)

fori =j, k, m, andn

A' =|<4>+¢i)c. (wwgc, 1, [<¢+4>i) - <w+wg1c| = 0 <4)

fori =j, k, m, andp

Each resultant can be expanded into a polynomial in the two
variables ¢t and wt. (Refer to Table I for the expansion of the
resultant A.) The resulting polynomials can each be factored into
a quadratic expression in (lit with coefficients that are quadratic

expressions in wt which have constant coefficients. Thus

10
“We; v.8 (14> aw) Cilia

-(i¢is’ w ’ 1’ (¢i-wi)c| + i¢is’ ‘1” ’ 1’ (¢i-¢i)siiwt

ic is

2
+i¢is' 9”- 1 1' (ti'h’sll‘it

1C

15 IS

2
+[(i¢is' W. ' 1' (chi-(1195' - i¢ic' W. ' 1' (¢i-wi)ci)wt

+(|¢is' W. ' 1' (¢i-¢i)ci + l¢ic' wic' 1’ (¢i-wi)cl

IS

+14%. 10. . 1. ”(Viki - ms. w. . 1. wi-wgslwt

IS 1C

- il¢is' W. ' 1' (¢l-¢/l)c I + i¢ic' wic' 1' (¢i-wi)s|l]¢t

1C

2
' |¢ic' ‘0' ' 1’ (¢i-wi)siwt

IS

+(l41. .

1C

wlC’ 1’ (¢i-¢i)si - |¢ic'wis' 1' (¢i-wi)cilwt
+ i¢ic‘ wic' 1’ (¢i-wi)ci : o (5)

fori=j, k, m, and ii

If in equation (5) the following substitutions are made

61 zi¢is' wis' 1’ (¢i-wi)ci
52 :|¢is’ wic' 1’ (¢i-wi)c|
53 =|¢is' wis’ 1’ (¢i-wi)si
54 Zi¢is’ wic' 1’ (q’i’wi) si fori =j, k. m, andn (5a)
55““!4’151013'11‘4’W’sl
56 Id” «lie. 1. (¢.- 1/1) |
67: chic' his. 1. (¢.-1/J.)C|
68: |¢ic' l/Jic. 1. (43- W1) cl

the result will be

2 2.
[filwt " (52+63)1//t+64] (pt
2 .
+[(63-67)(0t + (61+68+65-64)¢ t -(6 2t+66)]q>

Z
—65(,[/t+ (6 6 -67)¢1t + 68: 0

If in equation (5b) the following substitutions are made

a2 = 61, a1 = 63-67, a0 = -65
{32 .. -(62+6) 51 = 61+68+65-64, so = 66-67
V2 ___ 64' Y1 : ’(62+66)’ Yo : 68

the result will be
(aw2+w+ 1¢2+<w2+5w+1¢+aw2+w+
2t 2t Y.Zt alt' lt V1 t 0t 0t Y

Likewi s e

A. = [l¢is' W.

is'

2
1' (4’1 -wi)c iwt

-(|¢. ’w' ' 1' (¢i-wi)c'+i¢is’w

15 1C

1. (4., w )Snw

is

+ |<1>. 1. («pi-wig ”if

is' wiC'

+[(i¢is' wis' 1’ (¢i-wi)sl-|¢ic

is'

1' (¢i—¢i)c i )th

+”(pie wis’ 1' (¢i-wi)ci+i¢ic’ wic' 1’ (¢i-wi)c|

+g¢. .w. . 1. (¢i-wi)C| ‘li’is' we. 1. «pi-wig; '

1C 1C 1

-{{¢. . w. . 1. (abi~wi)s[+|¢ic.w . (¢. «M M]

IS IS

2
-i¢ic' ‘0' ' 1' (¢i-wi)si¢t

o ’
1C

+l¢ ! W 1 1: (¢l-wl)Ci :0

1C 1C

fori=j, k, m, andp

1, (¢i-Wi)si-I¢1C' wig, 1. “Pi-$9.1M

11

(5b)

(So)

(5d)

(6)

Also if in equation (6) the following substitutions are made

6'1 :|¢is
6'2 :l‘i’is
5'3 :l¢is
6'4 zi¢is
6'5 :|¢ic'
6'6 :i¢ic'
5'7 :|¢ic'
6'8 :|¢ic

the result will be

[ 6'lwt

2

. wis'
. V1. .
ic
. W.
is
. (1/.
1c

S

is'

ic

IS,

see

1C

1.

l.

l.

. (qt-W.)
1 1

(¢i-wi)ci
wwaJ

. (¢i-I//i)s|

. (43-15151

. (¢.—¢/.) ;
1 1 S

A

.(%%%%|

(¢i-wi)ci

l l l 2
‘ (62+63)Wt+ 641¢t

for i =j,

k. m, and p

I Z I I l I I I
+[(63- 67wt + (61+ 68+ 6 - 64wt- (62+ 66)]q>t

- 6i‘a‘thrwé'

5

5' + 6' =
7M 0

8

If in equation (6b) the following substitutions are made

l
‘'2

5'2

l
Y2

= I
61.

-(6' +6
2

= I
64.

the result will be

I 2 I I 2 I 2 I I
(a w + {5 wt +\(2)<1>t Halal!t +fi1¢t+vll¢t+a

2 t

2

').

7

6'+6'
1 8

II

6' - 6'.
3

- I I
(62+ 6

I_I
+65 64,

6)“

l:_6I
“o 5

I: I_ I

‘30 66 67
I: I

Yo 68
'w2+fi'w+v'=
0t 0t 0

Now if in equation (5d) the following substitutions are made

X =aw2+fiw+y forr=0,l,and2
r rt rt r

then equation (5d) may be put in the form

12

(6a)

(6b)

(6C)

(6d)

13

22: r
X 4) =0 (7)
O I‘. 1:

And if in equation (6d) the following substitutions are made

2
I -_—_ I I + I :
X arwt +{3rI/Jt yr forr 0,1, andZ

r

then equation (6d) may be put in the form

25X; <I>tr = o (8)

The left hand members of equations (7) and (8) are quadratic
expressions in the variable (pt with quadratic coefficients in the variable

([11: with constant coefficients, and if they are to have a common root for

(Pt, then their resultant must vanish, or

 

. 0 X', 0
X0' 0
I I
X1, X0, X1, X0 _
- ‘ “0 (9)
l l
X2, X1. X2, X1
X , , X'
0 , 2 0

 

Expanding the left hand side of equation (9) into a polynomial in

wt leaves 8 r
glen/7t = o (10)

Values of the coefficients Cr in equation (10) are given in Table II as
functions of the coefficients in the quadratic terms of the determinant
in equation (9). It is not necessary to generate all the coefficients Cr in
equation (10) for it can be shown that five of the roots of equation (10)
may be regarded as known. Therefore, equation (10) may be reduced

to the following cubic equation in the unknown (0t.

3
EA (or = o
r t
o
where
A0 : "Co/“1“2H3C8
C6
A1 2'5; ‘ (”1”? “1“3+ “2%) ‘1
C
+( + + )(—7 + + + )
“1 ”2 “3 c8 ”1 ”2 “3
A2 = C7/C8+ “1+ p2+ “3
A :
3 1
here

”1 = -[(¢/J.+I1/k)/th

I12 - -[(Wj + Wml/th

'[W/k +wmI/2]t

Also a somewhat less accurate calculation gives

c
A1:-__l—E—+AO(-_l—+'~J—+i)
”1“2“3 '8 “1 “L2 ”3

That is, less accurate in the sense that it tends to base the cubic

equation on the known roots rather than on the coefficients of the

eighth degree polynomial.

Equation (11b) gives three of the five known roots and the

remaining two are

II
+
H‘

“4
PS: -i

14

(11)

(11a)

(11b)

(11c)

15
Since ”5 = ~94. they do not appear in equations (1 la).
Each real root of equation (11) will provide a value for the starting

angle (0. Consequently, if all the roots are real. the starting angle VJ

will have three values.

When a value of II! has been determined, the two quadratic equations
(7) and (8) may be evaluated and solved simultaneously for ot, giving

I _ I
(p _ szo X0x2

_ (12)
I _ l
t xlx.Z xle

Thus, a pair of starting angles 4’ and W are determined.
It remains to be shown here that pl. Iiz. p3, H4- and us actually

are roots of equation (10).

The roots pl. I12, and u3 may be treated collectively for they are

similar.

When equation (12) is evaluated for “’1; = pl, the result will be

(.+
49 ._[¢ ¢k)]
t Z t

Therefore the pair of starting angles in this case is simply

(¢. + 4’1.)

4>=- 2

(w. + wk)
#1 : _ __.L2___
The insertion of this pair of starting angles in each determinant

of equations (3) and (4) will make the jth and kth row of each determinant

identical and each determinant will vanish as required.

16

Thus, p1, H2: and II3 are roots of equation (10).
That u4 and us are roots of equation (10) can be verified easily

by inserting the values of

¢t=1

wt=1
or

¢t=-1

«If-1

in equation (5b) and (6b). Either pair of values will make the left
side members of both equations (5b) and (6b) vanish.

Thus, and (is are roots of equation (10).

“4

The coefficients ar, fir, and yr for r = O, l, and 2 may be calcu-
lated in an alternate way. It can be shown that

a2 = --r1-'rZ-'r3-‘r4

= '2[n4tT 2' (94 ' n4)tT4]

y2=-Tl+rz-T3+‘r4

a =—2[9 'r +(9

4t 1 4 ” "4% T4] (13)

{31 = -4‘r4

:_ 6 .. 6 -
Yi 2[ 451 (4 "4)IT4]

a=T-T-T+T

I3 =-2[n 'r +(94-n4)

4t 2 9‘4]

T+T-T-T

here

-I
ll

17

1 "94¢ {(nzc'"1c) [(93'"3)c ' (91'n1)c]

'("3c ' "1C)[(92'n2)c ' (91'"1)c]}

-|
I

”(93c ' 61c) [(92'"2)c ' (ei'"1)c]}

2 _ "4c{(92c”91c)[(93'"3)c ' (91’"1)c]

(13a)
73 = ' {(91'"1)c (92cn3c ' 93c’72c)
-(92-n2)c (91"3c ' e3.2771c
+ (e3'n3)c (elanC ' 92871.9}
“'4 = (94474); [(92c'91c)(”3c’nic) ' (936913 ("zc'rhcn
where

91%” +¢k -¢m-¢n) n1 Jaw. wk ~10 -wn)
ez=—;(¢j -<I>k+¢m-¢n) n2=31<w -wk+w won)

1 1 (13b)
93=E(¢j-¢k-¢m+¢n) n3=3wJ-wk wmwn)
9,, =31<¢.+¢k +<I>m+¢n) n4 =§<w wk +wm+wnI

The coefficients a'
‘ r

,B', andy'r forr =0, l, andZmaybe
r

calculated by using (13), (13a), and a modified equation (13b). The

modification of equations (13b) being the replacement of on and (fin by

(pp and (0p.

l8

5. Determination of the Link Ratios. For every pair of starting
angles the position angles Qi and Ipi for i = l, 2, 3, 4, and 5 may be
evaluated by equations (2d) and (2e) respectively. When these position
angles are substituted in equation (2), there will exist five equations

in the three unknowns R1, R2, and R3. Any three of the five equations

may be solved simultaneously for the values of R1. R2, and R3;

however. it is better to solve all five equations simultaneously in
order to determine the accuracy of the calculation. When R1, and R‘2
have been determined, the link ratios Ll/L4 and L3/L4 may be

evaluated from equations (2a) and (2b) respectively. After these two

link ratios and R3 are known, the link ratio LZ/L

4 may be obtained

from equation (2c).

CHAPTER II

CALCULATIONS

1.. Solved Problems. In order to demonstrate the solution of the

 

five point four bar linkage problem given in articles (1) through (5),
two examples will be given here.

(1) Let f of equation (1),
(p' : f(¢n)
i i

be defined as a continuous function of chi, and suppose that
equation (1) is satisfied by the following discrete pairs of angles

which are also to be the displacement angles of a four bar

linkage.
4’1 = 05° 00' 00", “”1 = 04° 00. 00..
$2 = 15° 00' oo“, (02 216° 00. 00..
.¢3 = 25° 00' 00", (03 = 330 20. 00..
<I>4 = 35° 00' 00", .774 .—_ 56° 00. 00..
<I>5 = 45° 00' 00", (p5 .—. 84° 00. 00..

The problem here is to determine the link ratios and the starting

angles in the design of the four bar linkage. It follows that

23

(1)

and

-6- -9- -6- -6~ ~9- -G- '6- -6- ~6- ~6-
rh» L» N H U1 I!) W N F"
U) (D U) (D O O O O 0
II

Ln
0)

Froni

.99619470,

.96592583,

I

w
.90630779, w
.81915204, w
w

.70710678,

.08715574, w

(cpl-101)
(«12-sz
(cg-$3)
«14-114)
HIS-$5)

ls
.25881905, W

25
.42261826, w

35
.57357644, $45
.70710678, $53

equations (5a) and

01° oo'oo“

-01° oo'oo"

- 08° zo'oo“

21° 00' 00"

— 39° oo'oo"

=.99756405,
=.96126l70,
=.83548781,
=.559l9290,
=.10452846,
=.06975647.
==.27563736.
=.54950898,
=.82903757,

=.99452190,

(6a)

(<1 1' ”1’.
<¢2- 1021C
(<1>3- 1031C
(¢4- 1041C
(<15— wSIC
(<1 1- «0118
<¢2- «p218
(<13- I313
<¢4- 12,13

(<15— «ISIS

.99984770
. 99984770
.98944164
.93358043
.77714596
.01745241
.01745241
.14493186
.35836795

- .62932039

24

25

61 = -. 56650938 x 10'3, 81 = -. 26360800 x 10'2
62 = +. 43738916 x 103, 6'2 =+. 21229813 x10.2
63 = 83669494 x10-3, 6'3 = -. 39906703 x 10’2
(54 = -. 10916537 x 10'2, 6; = -. 41711880 x10”2
65 = +. 10695348 x 10°, 6; = +. 60278368 x 10'3
6(3 = +. 11983010 x10-3, 6'6 = +. 45634663 x 10'3
67 = -. 12389769 x10-3, 6'7 = -. 55927743 x 10'3
68 = —. 37663543 x 10"}, 6'8 = -. 19376427 x 10'3

From equations (5c) and (6c)

.56650938 x 10'3, a1 = -. 71279725 x 10‘3, a0 = -.10695348 x10.-3
—3 -3 -3
. 39930578 x 10 , 81 = +. 59443422 x 10 , I30 = +. 24372779 x10
-2 —3 -4
. 10916537 x 10 , v1 = -. 55721926 x 10 , v0 = -. 37663543 x 10
-2 -2 ~3
.26360800 x 10 , a'l = -. 34313929 x 10 , ab = -. 60278368 x 10
- -2 -2
. 18676890 x 10 2, 8'1 = +. 19441274 x 10 , °0 = +. 10156241 x 10
-2 -2 -3
.41711880 x 10 , v'l = -. 25793279 x 10 , y'o = -. 19376427 x 10
From Table II
-14
C0 = -. 23571951 x 10
c6 = -. 24518283 x10.-14
-13
c7 = -. 21154334 x 10
-15
c = -. 52287725 x 10

26

From equations (11b)

111 = -. 17632698
H2 = -. 33783302
.13 = -. 45924395

From equations (11a)

A = +. 16479023 x 103
A = -. 35040734 x 102
A = +. 39484149 x 102
A

= +. 10000000 x 10

Thus,equation (11) is

3

wt + 39. 484149 wtz - 35. 040734 0t + 164. 79023 = 0

The above cubic equation has only one real root which is

From equation (12)

wt = -40. 451105
77 = -88° 35' 02"
cpt = -1. 0799791
4> = -47° 12' 07"

From equations (2d) and (2e)

41 = -42°12'07", 01 = .—84° 35' 02"

42 = -32°12'07", 02 = -72° 35' 02"

43 = -22°12'07", 03 = -55°15'02"

44 = —12°12'07", 04 = —32° 35' 02"

45 = -02°12'07", 05 = -04° 35' 02"
Then
41C = .74078180, 01C =.09438825, (41 - 01)c =.7386678O
42C =.84617508, 02C = . 29930911, (42 - IIIZIC = .76174251
43C = .92585776, 03C = . 56998880, (43 - 03)C = . 83820818
44C = .97740872, 014C = . 84260386, (44 - 4’44 = .93739179
45C = .99926161, 05C = .99680139, (45 - 415% = .99913598

Evaluating equation (2) for i =1, 2, 3, 4, and 5 gives

. 74078180 R1 + . 09438825 R2 + R3 = . 73866780
. 84617508 R1+ . 29930911 R2 + R3 = . 76174251
. 92585776 R1 + . 56998880 R2 + R3 = . 83820818
. 97740872 R1 + . 84260386 R2 + R3 = . 93739179

= .99913598

.99926161R1+ .99680139 R‘2 + R3

Subtracting the first equation from the last four and dividing by
the coefficient of R1 in each of the remaining four equations

leaves

28

R1+1.944344648 R2 = . 21893910
R + 2. 569758654 R = . 53783428
1 2
(b)
R1 + 3.162005447 R2 = . 83981987
2 1. 00769255

R1 + 3. 491232448 R2

Subtracting the first equation from the last three and dividing
by the coefficient of R2 in each of the remaining three equa-

tions leaves

R2 = . 50989612
R2 = . 50989633 (b1)
R2 = . 50989700
Taking
R2 = . 5098965 (c)

and substituting this value in equations (b) and solving each of

the four equations for R determines

1
R1 = -. 772475430
R1 = -. 772475668 (c1)
R1 = -. 772475637
R1 = -. 772474652
Taking
R = —. 772475 ((1)

l

and substituting this value and the value of R2 from (c) into

29

the five equations (a), and solving each of the five equations

for R3 determines

R = 1. 26277498

3
R3 = 1. 26277493
R3 = 1. 26277486 (<11)
R3 = 1. 26277483
R3 = 1. 26277504
Evidently
R3 = 1. 26277 (e)

Substituting the value of R from (d) into equation (2a) gives

1
1.1/L4 = -1. 29454 (f)

Substituting the value of R from (c) into equation (2b) gives

2

L3/L4 = -1. 96118 (g)

Substituting the values of R from (e). of Ll/L4 from (f), and

3
of L3/L4 from (g) into equation (2c), and solving for the link
ratio LZ/L4 gives

: . l
LZ/L4 33 787

Thus the required link ratios are

Ll/L4 = 1. 29454
1.2/1.4 = . 331787
1.3/1.4 = 1.96118

30

where the negative signs of the values of Ll/L4 and L3/L4 have
been removed. The removal of a negative sign from a link
ratio may always be accomplished by the addition of 1800 to the
starting angle associated with the link ratio.

The above link ratios and the two starting angles

132° 47' 53"

¢

910 241 58”

10

define the four bar linkage for the given displacement angles.
The mechanism is shown in Fig. (2) in its first precision
position in full lines and in each of its remaining four positions

in dashed lines.

(2) As in example (1), assume that a function f has been given as

w, = £14.)
1
and suppose f to be satisfied by the following discrete pairs of

angles which are also to be the displacement angles of a four bar

linkage function generator of f

<71 = 01° 03' 54. 85", 01 = 00° 06' 57. 21"
4.2 :120 37' 45- 90"» 102 = 04° 43' 51. 63"
¢3 = 360 53' 38. 30", (p3 : 23° 371 18. 26"
«P4 = 67° 02' 48. 44", $4 = 57° 52. 8. 28..
4’5 = 87° 19' 26.95", $5 : 86° 00. 58. 36..

Again the problem is to determine the link ratios and the

starting angles of the four bar linkage.*

and

-e -e -e -e -e -e 43 4+ e» -e
vb U) N t-* U1 1% L» N *-‘
(D 01 (I) In D O O O O

U1
(I)

It follow 5 that

(41-01)
(42-42)
(¢3-W3)
(44-44)
(<PS-W5)

. 99982717, W
1c

.97580464, W
2c

.79974782, W
3c

.38997930, W
4c

.04668573, W
5c

.01859080, W
ls

.21864427, W
23

.60033609, W
35

.92082362, W
45
.99890965, W55

00° 56' 57. 64"

07° 53' 54. 27"

= 13° 16' 20. 04"

= 09° 10' 40.16"

= 01° 18' 28. 59"

= . 99999796.
= . 99659290.
= .91621077,
= . 53185733,
= . 06947422,
= . 00202269,
= .08247787,
= . 40069668,
= . 84683397,

=.99758375,

were taken from Freudenstein (2).

(41-011,:
(42-wZIC
(43-1031,:
014-10,)c
(cps-051C
(41-0118
(42-0215
(43-4315
144-0415

145-0515

*These points will satisfy the requirements
generator of the form y = x1- 5 with suitable scale

.99986274

.99051328

.97329025

.98719808

. 99973946

.01656843

.13741703

.22957809

.15949908

.02282591

for a function
factors and

31

From equation (53.) and (6a)

0‘!
ll

0‘!
II
I

-4
6 = +. 95820540 x10 .

+. 91278590 x10- .

. 38032454 x10- ,

3 I
°-1
3
I
°2
6|
3

. 65941582 x 10'2, 6'

.14316076 x 10'

32

2

-3
. 56289109 x10

.26574678 x10-2

1

: _ : -. 4 4 -
64 4 166 32 6 x10
-2 -1
65 = +. 37584876 x 10 , 6'5 = +. 10650043 x 10
66 = +. 33814141 x 10'2, 6'6 = +. 87864628 x10-2
67 = -. 10868160 x 10'4, 617 = -. 28898921 x 10'3
68 = -. 17119667 x10-3, 623 = -. 48642779 x 10'3
From equations (5c) and (6c)
-3 -3 -2
+.91278590 x 10 , a1 = +.10668870 x 10 , a0 = -. 37584876 x 10
+. 28450400 x 10'3, 61 = +. 11094235 x 10'1, 80 = +. 33922823 x 10'‘2
.65941582 x 10'7". yl = -. 30010895 x10-2, v0 = -. 17119667 x10.3
-2 _ -2 -1
+. 14316076 x 10 , a'l — -. 23684787 x 10 , ab = -. 10650043 x 10
+. 32203589 x 10°, 8‘, = +. 28238469 x 10‘1, °b = +. 90754520 x 10’2
. 16643246 x 10'1, y'l = -. 82235717 x 10’7”, 9'0 = -. 48642779 x 10"3
From Table II
-13
c0 = -. 94005139 x 10
_ -10
c6 — +. 67677143 x 10
-10
c7 = -. 26412843 x 10
-11
c = -. 43945779 x 10

From equation (5a) and (6a)

.1.

+

+

0'!
ll

.65941582 x10-

.33814141 x 10'

.17119667 x10-

3

+. 91278590 x 10' ,

2

2

3

From equations (5c) and (6c)

.91278590 x10-3,

. 28450400 x 10'3,

.65941582 x 10'2.

.14316076 x 10'2,

-2
.32203589 x 10 ,
.16643246 x 10'1,

From Table II

. 38032454 x 10'3,

-4
. 95820540 x 10 .

-2
. 37584876 x10 .

O

. 10868160 x 10'4.

5.
5.
5.
5.
5.
5.
5.

I
68

. 28898921

. 48642779

= +.10668870 x 10'3,

+.11094235 x 10'1,

.30010895 x 10' .

.23684787 x 10' ,

2

2

+. 28238469 x 10'1,

-. 94005139 x 10'
+. 67677143 x 10'
-. 26412843 x 10'

-. 43945779 x 10'

.82235717 x10-

2

13

10

10

11

. 14316076 x
. 56289109 x
. 26574678 x
. 16643246 x
. 10650043 x

. 87864628 x

X

X

32

-. 37584876 x 10'2

+. 33922823 x 10'

2

-. 17119667 x10.-3

-.10650043 x10”l

+. 90754520 x 10'2

-. 48642779 x 10'

3

From equations (11b)

From equation (11a)

A :

3
Thus, equation (11) is

3

11> :> 3"
II

= -. 04232242

-. 21016469

-. 25260047

+

. 95207252 x10

. 19253445 x 10‘2

+

. 55052349 x10

+. 10000000 x 10

wt + 5. 5052349 Wtz - 19. 253445 wt + 9. 5207252 = 0

The above cubic equation has the following real roots

From equation (12),

Therefore, the following

= -8. 0454164
= +1. 9256499

= +0. 6145314

= -1. 6229270
= +6. 6172438

= -0. 1092704

pairs of starting angles will exist.

33

Thus, there are three

Case I

¢7= 31° 34119.31"

¢:=-O6O 14'09.58”.

5
ll

62° 33'24.89"

¢== 81° 24'23.33",

¢.:=-82° 54'53.26”

.9.
ll

-58° 21'35.43",

cases to be considered

When the first set of starting angles is inserted in

equations (2d) and (2e), the result will be

191
II

'64
II

16‘
ll

'61
ll

191
ll

. 99593053.
. 99378071.
. 86022648,
. 48769519.

. 15491457.

-05° 10'14.73", 01
0 .

06 23'36.32", 02
O

30 39'28.72”, 03
O

60 48'38.86”, 04

81° 05' 17. 37", 05

41C = .85092186.

42C = .80589688.
43C = .57080286.
44C = .00975628,
45C =-.46311416,

31°.41'16.52"

36° 18110.94"

55° 11137.57"

89° 26'27.59"

117° 35'17.67“

(°1"1’1)c
(@2'°2h:
(93-43%
(34%).;

(Q5-¢5)c

II

. 80011745

. 86681283

. 90970204

. 87773051

. 80385600

34

35

Evaluating equation (2) for i = l, 2, 3, 4, and 5 gives

.99593053R +.85092186R +R =.80011745

1 2 3

.99378071R1 + . 08589688 11‘2 + R3 = . 86681283

.86022648 R1 + . 57080286 112 + R3 = .90970204 (a)
.48769519 R1 + . 00975628 R2 + R3 = . 87773051

. 15491457 R1-.46311416 R2 + R3 = . 80385600

Subtracting the first equation from the last four and dividing
by the coefficient of R1 in each of the remaining four equations

leaves

R1 + 20. 94360458 R2. = -31. 02370431
R1 + 2. 06419042 R2 = - . 80752630
R1 + 1. 65507101 R2 = - . 15271087 (b)
R1 + 1. 56243886 R2 = - . 00444528

Subtracting the first equation from the last three, and dividing

by the coefficient of R2 in each of the remaining three equations

leaves
R2 = -1. 60048282
R = -1. 60048421 (b)
2 1
R2 = -1. 60048469
Taking
R = 1. 60048 (c)

‘36

and substituting this value in equations (b) and solving each of

the four equations for R determines

1

R1 = 2. 4961160
R1 = 2. 4961692
(c1)
R1 = 2. 4961972.
R1 = 2. 4962069
Taking
R = 2. 49617 (d)

1
and substituting this value and the value of R2 from (c) into

equation (a). and solving each of the five equations for R

3
determines
R3 = -. 32401103
R3 = -. 32401092
R = -. 32401093 ((1)
3 1
R3 = -. 32402486
R3 = -. 32404205
Evidently
R3 = -.324019 (e)

Substituting the value of R from ((1) into equation (2a) gives

1

Ll/L4 = .400614 (f)

37

Substituting the value of R from (c) into equation (2b) gives

2

L3/L4 = . 624813 (g)

Substituting the value of R from (e), of Ll/L4 from (f), and

3
of L3/L4 from (g), into equation (2c), and solving for the link

ratio L3/L4 gives
1.2/1.4 = 1. 30885
Case II

When the second set of starting angles is inserted in

equation (2d) and (2e), the result will be

41 = 82° 28' 18.18", 4.1 = 62° 40' 22.10"
0 O
42 = 94 02' 09. 23". 11.2 = 67 17' 16. 52"
O O
43 = 118 18' 01. 53", 11.3 = 86 10' 43.15"
0 O
44 = 148 27' 11. 77", 4.4 = 120 25133.17"
45 = 168° 43' 50. 28", 1115 = 148° 34' 23. 25"
Then
4 - .1 1 1 . = . , - =,.
1C 3 0 559 011C 45907127 (41 011% 94028721
4 =-. 4 = . 1 1, - = . 4
2C 070381 6, 42C 386 005 (42 ¢2)c 89299 85
4 =-.4444, =. 4, 4- =.
3C 7 09 7 43C 0666 565 ( 3 ¢3IC 84691967
44C = -. 85221373, 414C = -. 50642333, (44-44% = .88272307
4 =-. 80128, =-.833 , 4- =. 4 4
5C 9 7 9 45C 5 0632 ( 5 “’51.: 9387 88

38

Evaluating equation (2) for i = l, 2. 3, 4, and 5 gives

. 13101559 R1+.45907127 R2 + R3 = . 94088721
-. 07038146 R1 + . 38610051 R2 + R3 = .89299485
-. 47409474 R1 + . 06664565 R2 + R3 = .84691967 (a')
-.85221373 R1 - .50642333 R2 + R3 = .88272307
-. 98071922 Rl - .85330632 R2 + R3 = .93874884

Subtracting the first equation from the last four and dividing

by the coefficient of R in each of the remaining four equations

1

leaves

R1 + . 36232288 R2 = . 23780070

R1+ . 64851912 R2 = . 15528993
(b')

R1 + . 98196278 R2 = . 05915623

= . 00192345

Rl +1. 18047713 R2

Subtracting the first equation from the last three, and dividing
by the coefficient of R.2 in each of the remaining three equations

leaves

39

R2 = -. 28830138
R = -.2883o369 (b')
2 1
R2 = -. 28830413
Taking
1?»2 = -. 288303 (c')

and substituting this value in equations (b') and solving

each of the four equations for R1 determines
R = . 34225947
R = . 34225894

(Cl)
R = . 34225905

1
R1 = . 34225855
Taking
R1 = . 342259 (d')
and substituting this value and the value of R2 from (c')

into equation (a'), and solving each of the five equations

for R3 determine 5

40

R3 = 1. 02839757
R3 = 1. 02839748
R3 = 1. 02839700 (d'l)
R3 = 1. 02839752
R3 = 1. 02839807
Evidently
R3 = 1. 02840 (e')

Substituting the value of R1 from (d') into equation (2a) gives

Ll/L4 = 2. 92176 (f')

Substituting the value of R from (0') into equation (2b) gives

2
L3/L4 = 3. 46857 (g')

Substituting the value of R from (e'), of 1.1/1.4 from (f'), and

3
of L3/L4 from (g'), into equation (2c), and solving for the link
ratio L3/L4 gives

1.2/1.4 =.850513

Case III
When the third set of starting angles is inserted in

equations (2d) and (2e), the result will be

Then

Q1c

4
2c

3c

<I>
4c

4
5c

Evaluating equation (2) for i 2

I61
I

'91
l

161
ll

164
II

I61
ll

.54031954,
.69803514,
.93063572,

. 98852826,

87492152,

. 54031954 R

1

. 69803514 R1

. 93063572 R

l

. 98852826 R1

. 874921.52 R1

+ . 12535223 R

+ . 20477305 R

+

+ . 90596948 R

+

--57° 17140.
--45° 43149.
—21° 27'57.
08° 41113.

28° 57'51.

LIJ5c

. 51064713 R

.99853533 R2

58", 1111 =

53", 1112 =

13", 1113 =

01", 1114 =

52", 1115 =
.12535223,
. 20477305,
.51064713,
. 90596948,
. 99853533,

2, 3, 4, and

2 + R3 =

2 + R3 =

2 + R3 =

2 + R3 =

+ R3 =

-82
-78
-59
-25

03°

(@1-¢l)c
(QZ-¢Z)C
(935-1113)C
(83,4115!)C

(§5-¢5)c

5 gives

. 90255299

. 84382853

. 78986411

. 83163640

. 89984047

47' 56.

11' 01.

17' 35.

02' 44.

06' 05.

05H

63H

00”

98H

10”

. 90255299

. 84382853

. 78986411

. 83163660

. 89984047

Subtracting the first equation from the last four and dividing

by the coefficient of R1 in each of the remaining four equations

leaves

41

(a")

42

R11 + .50356984 R2 = -. 37234402

R + .98713535 R = -.28871178
1 2 II
(b 1
R1 + 1. 74163780 R2 = -. 15822180
R1 + 2. 60961725 R2 = -. 00810671

Subtracting the first equation from the three, and dividing by

the coefficient of R2 in each of the remaining three equations

leaves
R2 = . 17294914
R2 = . 17294868 (b'l')
R2 = . 17294829
Taking
R2 = . 172949 (c")

and substituting this value in equations (6") and solving each of the

four equations for R determines

1
R1 = -. 45943592
R = -.45943485
1 (6")
R1 = -. 45943632 1
R1 = —. 45943740
Taking
R1 = -. 459436 (d")

and substituting this value and the value of R2 from (c") into

equation (a"), and solving each of the five equations for R3

determines

R =1.12911570

3
R3 =1.12911571
R3 =1.12911554
R3 =1.12911555
R3 = 1. 12911522
Evidently
R3 = 1.129116

Substituting the value of R from (d") into equation (2a)

1
gives

Ll/L4 = -2.17658

Substituting the value of R from (c") into equation (2b)

2
gives

1.3/1.4 = -5. 78205
Substituting the value of R3 from (e"), of Ll/L4 from (f").

and of L3/L4 from (g"), into equation (2c), and solving for

the value of the link ratio L3/L4 gives

1.2/1.4 = 3. 27865

Thus, there are the following three solutions to the problem.

43

(«1'10

(e")

(f")

(g")

44

Case I
Ll/L4 = . 400614
1.2/1.4 = 1.30885
L3/L = .624813
4 = 31° 34' 19. 31"
4 = ~06O 14' 09. 58"

This mechanism is shown in Fig. (3).

Case II
1.1/1.4 = 2.92176
LZ/L4 = .850513
1.3/1.4 = 3. 4685?
w = 62° 33' 45. 51"
O
4 = 81 24' 27.72"

This mechanism is shown in Fig. (4).

Case III
1.1/1.4 = 2.17658
1.2/L4 = 3.27865
1.3/1.4 = 5.78205
0.: 97° 05' 6.74"
<I> = 121° 38' 24. 57"

I

This mechanism is shown in Fig. (5).

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49

2. Problem Discussion. The values of the trigonometric function

 

of the displacement angles in Example (1) and (2) were taken from
Peters (3) wherein trigonometric values are given to eight places as
functions of angles given in degrees, minutes, and seconds. Therefore,
the trigonometric values of the displacement angles of Example (1) are
accurate to the eight places given since no interpolation is required to
obtain them. The values of the trigonometric functions of the displace-
ment angles given in Example (2) are not accurate to the eight places
given because they are obtained by linear interpolation in Peters (3).
The values of the starting angle 1// for each example were also
determined via the generation and solution of equation (10) with the aid

of a digital computer. Therefore, the following data are available.

For Example (1)

The coefficients in equation (10) are

c0 = -.23571951x10-14
-13
c1: -. 24977260 x10
-13
c2 = -. 81377704 x10
-13
c3 = -.99426682x10
-13
c4 = -. 80949468 x 10
-13
c5 = -.95603762 x10
c6 = —. 24518283 x10‘14
-13
c7 = —. 21154334 x10
-15
c = -. 52287725 x10

The roots of equation (10) are

”I = -. 17632691
112 = -. 33783343
113 = -. 45924364
04 = +1. OOOOOOOi
115 = -1. OOOOOOOi
116 = -40. 451105
p7 = +. 48347782 + 1. 95960761
p8 = +. 48347782 - 1. 9596076i

For Example (2)

The coefficients in equation (10) are

c0 = -. 94005140 x10-13
-11
c1 = -. 28505340 x 10
Cz = -. 15132305 x10-10
-11
c3 = -. 37228276 x 10
-10
c4 = +. 57033385 x 10
-10
c5 = -. 27285139 x 10
-10
c6 = +. 67677143 x 10
-10
c7 = -. 26412843 x 10
-11
c = -. 43945779 x 10

51

The roots of equation (10) are

‘ . 04232188

1:
g—a
l
I

pa = . 21016715

113 = . 25259804

94 = +. 999999991
.15 = -. 999999991

“7 = +1. 9256406

- . 80454201

Although the roots of equation (10) as given by equation (11b) are
theoretically correct, they become approximately correct when approxi-
mations are made for the trigonometric functions of the displacement
angles. The error involved in reducing equation (10) to the cubic
equation (11) is very small when the reduction is accomplished via
equations (11a) and (11b). This can be seen when each solution of
equation (10) given above is compared to each solution of the cubic
equation (11) given in the examples. It is to be noted that the roots of
equation (10) pertain to the solution of the mathematical problem estab-
lished by giving certain trigonometric functions certain values, and it
is tacitly assumed that all trigonometric relations involved will be

satisfied by these values as they were for the trigonometric functions.

52

The approximations referred to above are to be understood as
mathematical approximations and are of academic interest only. From
a practical point of view the accuracy of the link ratios determined by
the use of eight place trigonometric values is far beyond that which is
normally required. If more accuracy is desired in the calculation, the
values of the trigonometric functions can be calculated as accurately as
desired by the use of series expansions.

The accuracy of the present calculations can be deduced from
the values of the Rj's given in (b1), (c1), and (d1) in Example (1), and

in (bi), (Ci), (di), (b'l'), etc. in Example (2).

53

CHAPTER III

SPECIAL CASES

1. Precision Derivatives.* In the design of a four bar linkage to be

 

used as a generator of an arbitrary function as f in equation (1), it may be
desirable to have a precision derivative at a precision point. That is,
it may be desirable to have the rate of displacement of angle (Di with
respect to the rate of displacement of angle ¢i equal to the derivative
of f with respect to ¢i at a precision point. The inclusion of a precision
derivative at a precision point, which will be seen later, can only be
accomplished at the expense of a precision point in the four bar linkage
generator design.

It is not difficult to adapt the five precision point design calcu-
lation to accept precision derivatives. To this end, there are the
following considerations.

If the function f has a derivative at a precision point, then

(111/.

1

(34>.

1

 

= 4'14.)
1
will exist at the point.

Equation (2) may be put in the form
R1(¢+¢i)c + R2(W+wi)c + R3 = [(440) +(<1>i - 4111C

*Refer to Hinkle (4) for a discussion concerning a precision
derivative.

54

Taking a derivative of the preceding equation with respect to chi, the

re sult will be

 

 

 

 

dwi dwi
Rl<4+¢i)s+nz<w+wi)s d¢i = 114-4) +14),- wins <1 - a?
or
dwi ' dwi
Rlgis + RZqus d4>i : (Qi-qjig (1 - d¢i ) (14)
where
1’. = ¢ + 9.
1 1
4Ji = W + 0,
dwi
d (,1 = 9141)

When evaluated at a precision point, equation (14) will be a
linear equation in R1 and R2, and may replace any precision point
equation (2) other than that of its) associated precision point in the
system of five simultaneous equations in the five precision point
design calculation.

An entry of equation (14) as one of the five simultaneous
equations will necessitate a modification of the determinants in
equations (5a) and (6a). In order to modify the determinants, suppose
that at a precision point q there is also to be a precision derivative.

Then equation (2) and equation (14) can be evaluated at the precision

point q providing two of the five simultaneous equations in the design

55

calculation. Suppose further that the rth precision point in the design
calculation be foregone in order to accept the slope precision equation (14).
The changes that are to be made in the determinants in equations (5a)

and (6a) as a result of the above modifications are in the rth rows, and
they are tabulated in Table III, page 57.

If a precision derivative is to be incorporated at each of two
precision points, then the procedure given above must be duplicated for
the second precision point.

In order not to disturb the procedure for the reduction of the
eighth degree equation (10) to the cubic equation (11), the following selec-
tion of the values of q and r should be made. If one precision derivative
is to be incorporated at one precision point, select the value of q from
the set j, k, and m and the value of r from the set n and p. If a precision
derivative is to be incorporated at each of two precision points, select
the values of the q's to have different values from the set j, k, and m
and the r's to have different values from the set n and p.

When the above changes have been made, no further changes in the
design calculation are necessary. Keeping in mind, of course, that the
link ratios will be determined from a set of five equations of type (2) and
(14) as required by the type of precision points that have been employed
in the design of the linkage.

Since each derivative of equation (2) will be a linear equation in R1
and R2, it is obvious that the calculation may be modified to accept a

number of higher order derivatives at each of two precision points.

56

2. Four Precision Point Design. It is obvious that a four precision

 

point design calculation is simply a part of the five precision point
design calculation regardless of the type of precision points employed
in the design.

In the four precision point design calculation one of the starting
angles is entirely arbitrary and the other starting angle may be deter-
mined from equation (7). Theleft hand member of equation (7) is a
quadratic expression in ¢t with coefficients that are quadratic expressions
in wt which have constant coefficients. A Therefore, if an arbitrary
value is assigned to one of the starting angles and its tangent value
inserted in the left hand member of equation (7). the result will be a
simple quadratic expression in the tangent of the other starting angle.
Thus, equation (7) will be reduced to a simple quadratic equation which
can readily be solved and the starting angle determined. If the solutions
of the reduced equation (7) are real values, then both values should be
considered for the starting angle. If the solutions are complex roots.
then another value of the arbitrary starting angle must be assigned and
the above procedure repeated.

When the starting angles have been determined. the link ratios
may be determined in the same manner as in the five point calculation;
however. there will be only four simultaneous equations involved in the

calculation.

57

TABLE III

PRECISION DERIVATIVE MODIFICATION

If the rth precision point equation (2) is replaced by the qth
precision derivative equation (14) in the set of five equations in the
design calculation, the following changes are to be made in the de-

terminants in equations (5a) and (6a).

 

 

Column Change To
(1) 4rc ¢qs
-¢
rs qc .
dxpi *
dwi
wrs -l//qc [E q
(3) 1 0
dwi
(4) (¢r-Wr)c (411'qu (1 - [dcpi lq)
d'll

i
(ctr-41.13 -(¢q-wq)c (1 - [34:101)

CW1. dW.
* [——1- represents 1
d¢. q d<1>i

 

evaluated at the qth point.

58

CHAPTER IV

CONCLUSION

1.. Five Precision Point Design Calculation. Since the determina-

 

tion of one of the starting angles depends upon the solution of a cubic
equation, there will always exist at least one pair of starting angles,
and there is a possibility of the existence of three pairs of starting
angles. It is unlikely that an equation of less than the third degree
exists for the determination of one of the starting angles because it is
shown in Example (2) that the existence of three pairs of starting
angles provides three different four bar linkages which are capable
of satisfying the same five precision points.

It is obvious that any equation developed for one starting angle
could likewise be developed for the other starting angle, and in
general, both starting angles possess the same characteristics.
Thus, the calculation is symmetrical with respect to the starting
angles.

The resultant in equation (3) or (4) can be expanded into a poly-
nomial in the two variables 0t and wt by the application of a double
Taylor's series expansion about the origin, and the resultant in
equation (9) may be expanded into equation (10) by the application of a

Taylor's series expansion about the origin.

59

Although the design calculation is rather lengthy, it is inherently
accurate because it contains no approximations, iterations, or graphi—
cal solutions.

2. Four Precision Point Design Calculation. When one of the starting

 

angles is assigned the value of zero, the four precision point design
calculation may be worked with sufficient accuracy in a reasonable time
by the use of logarithmic tables. There are an infinite number of
different four bar linkages that can be used to generate (approximately)
an arbitrary function given by four specified precision points.

3. Precision Derivatives. The theoretical consideration of the five

 

precision point design calculation can be used to formulate procedures
whereby a function generator can be designed with precision derivatives
at more than one precision point. These formulized procedures can be
obtained directly by modifications to five precision point design calcu-
lation without difficulty. It is also true that the design calculation can
be modified to accept higher order derivatives at different precision

points .

6O

BIBLIOGRAPHY

References Cited

 

(1) "Approximate Synthesis of Four-Bar Linkages" by Ferdinand
Freudenstein, Transactions of the American Society of

Mechanical Engineers, vol. 77, 1955, pp. 853-861.

 

(2) "Four-Bar Function Generators" by Ferdinand Freudenstein,

Machine Design, vol. 30, no. 24, 1958, pp. 119-123.

 

(3) ”Eight-Place Table of Trigonometric Functions for Every
Sexagesimal Second of the Quadrant" by J. Peters, Edwards
Brothers, Inc. , Ann Arbor, Michigan, 1943.

(4) "Kinematics of Machines" by Rolland T. Hinkle, Prentice-Hall,

Inc., Englewood Cliffs, N. J., 1960.

Re ferenc e Reading

 

(1) ”An Analytical Approach to the Design of Four-Link Mechan-

isms, " by Ferdinand Freudenstein, Transactions of the Ameri-

 

can Society of Mechanical Engineers, vol. 76, 1954, pp. 483-492.

 

(2) "Kinematic Analysis, " by Joseph Kaplan and Berthold Pollick,

Machine Design, vol. 26, no. 1, 1954, pp. 153-160.

 

(3) "Structural Error Analysis in Plane Kinematic Synthesis" by

Ferdinand Freudenstein, Transactions of the American Society

 

of Mechanical Engineers, vol. 81, 1959, pp. 15-22.

 

(4) ”Kinematics and Linkage Design" by A. S. Hall, Jr. , Prentice-

Hall, Inc., Englewood Cliffs, N. J., 1961.

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