4 Bar Mechanisms

8/11/2019 4 Bar Mechanisms

1/49

CHAPTER

5

AN INTRODUCTION

TO THE

SYNTHESIS

OF

FOUR BAR

MECHANISMS

5.1

Introduction

This

chapter

is

concerned

with

the synthesis or design

of

four

bar

mechanisms.

That

is, given

a desired

movement design the

four

bar

mechanism

that provides

that

movement. We

start this kinematic

synthesis with

a neecl

to

provide

some

motion,

not

a mechanism.

Our goal

is

to

find the mechanism that

meets

our needs.

This

is very

different

from

the

analysis tasks

discussed

in

the previous

three chapters

where

we

were given

the

mechanism,

and

asked

to

find

positions,

velocities

and

accelerations

of

parts

of the

mechanism.

There

are

several

stages

in

the general synthesis

or

design

process. We

are

starting

with

a need

to

provide

a motion. However,

considerable

effort may

have

been

made

by the designer

prior

to

this

stage

to define the

overall

system

in

which

a four

bar

mechanism

may be

only

one

part.

Given

a need to

provide

a

particular

motion,

it is

important

to consider

the

means

that

mav

be used

to

provide

that

motion.

Clearly,

a

four bar

mechanism

is not the

only

means

at

our

disposal

for

providing

a

specified

motion.

Recalling

the back

hoe

shovel

discussed

in

Chapter

1., the

four

bar

function

generator

served

to rotate

the

shovel.

Other

means could

have

been

used

for

that

task

including

elechic

or

hydraulic motors that produce

a

rotational

output

directly.

Using

such

clevices

would

eliminate

the need

for

a

four

bar

mechanism.

The

designer

had

to

choose

between

these

and

manv other options, but decided

after

consideration

of

multiple

factors

that

a

four

mechanism

driven

by a

hydraulic

cylinder

was

the

preferable

means

for

providing

the motion

of the

shovel.

Some

of

the

factors

which

entered

into

this

decision

making

process

may

have been

cost,

maintenance,

reliabilitv

and the

available

power sources.

Choosing

between

alternatives

is

an

important

parf

of the

engineering

design

process. In

kinematic

synthesis

this

particular

decision

making

process

is

sometimes

referred

to as type

synthesis

(ref).

While

this

has

been

a

somewhat

qualitative

process,

some relatively

recent

efforts

have

been

made

to

make

this process

more

quantitative.

For our purposes,

we

will

accept

that the

clesirecl

means of

providing

a

motion

is through a

four bar mechanism.

Thus,

we

will

concentrate

on

finding

the

four

bar mechanism

that suits

our

needs.

This

process

is

sometimes

referred to

as

climensional

synthesis (re0.

5.2

Classification

of

Synthesis

Tasks


2/49

In

Chapter

1

u.e saw

that

mechanisms may be designed

for different

purposes.

The

mechanism

that

drove the

back

hoe

shovel

was called

a

function

generator.

In

this

case

the

mechanism

is

designecl

to

provide

a

coordinated

motion

between

the

input

and

output

links.

This

motion

is coordinated

by

a

mathematical

function

of

one

variable.

The

mechanism

shown

in

Figure

5.1

is an

example

of

another

use

of a

function generator.

This mechanism

is

used

to control

the

air-fuel

mixfure

in

a

furnace.

In

this

case

it

was

important

to

maintain

a

linear

relationship

between

the

air

and

fuel

content

such

that when

the fuel

supply

decreased

toward zero

the

air

supply

also decreased

toward

zero. Thus,

this

mechanism

expresses

a

linear functional

relationship

with

a

zero

"y

intercept"

between

the rotation

of

the shafts

attached

to

the

input

and

output

links.

However,

it

was

also necessarv

to

provide

a

family

of straight

lines

through

the

origin

of the

fuel

air relationship to

account

for

different

operating

conditions.

To

provide

this linear

relationship,

over

a

range

of conditions,

this

particular

mechanism

was

made adjustable:

pneumatic

cylinders

in

the

coupler

and

output

links could

be

used to

change

the

link

lengths

to

maintain

the desired

functional

relationship.

Another example

of a

function

generator

is a

door

closer

(Figure

5.2). Here

the rotation

of the door

is

to

be controlled,

and

the door

is a

link

which

is

attached

to

the

frame.

In

each

of

these

examples

the

mechanism

is

designed

to

meet

is some

specified

relationship

between the rotation

of the

input and

output

links.

The

front

loader

discussed

in

Chapter

1

is

an

example

of a

motion

generating,

or

rigid

boclv guidance

mechanism.

This

mechanism

was designed

to produce

a

specified

rigid

bodv

motion

of the

coupler

link. In

some

automobiles

the

windshield

wiper is attached

to the

coupler

link

of

a four

bar

rigid

body

guidance

mechanism

(Figure

5.3).

Such

a mechanism

allows

the

wiper

to

clear

a larger

area

of

the

windshield

than

would

be possible

if

the

blacle

was

rigidly

attached

to

a single

rotating

arm.

In

each

of

these

motion

synthesis

mechanisms

the angular

rotation

of

the

coupler

as well

as

the

displacement

of a

point

on

the coupler

was

specified

bv the

designer.

Specif,ving these

three variables

fullv

locates

a

rigid

body

in

a

plane,

a,ncl

hence

the

name

rigid

body

guidance.

If

only the displacement

of a

point

on

the coupler

is

specified

in

a desigry

then

the

design

task

is

known

as

path

svnthesis.

Some

common

paths

for

which

mechanisms

have

been

designed

are straight

line

and circular

segments,

but

the

clesign

process

is

not

limited

to

these

paths.

Some well

known

straight

line

path

generating

mechanisms

are

shown

in

Figure

5.4.

It

is

important

to realize

that

manv

other types

of mechanical

svstems

can

produce

shaight line

motion

such

as

the.utu-follo*"i

t.

rack

and

pinion

shown

in

Figure

5.5.

One feafure that

distinguishes

four

bar

mechanisms

is that

the

driving

stroke

(the

specified path

of a

point on

the

coupler)

and

refurn

stroke

are

nof

in

general, the

same. This

is

a

useful

feafure

in

systems

such

as

the

film

advance

mechanism

shown

if

Figure

5.6.

The

point

P

on

the coupler

develops

straight

line

motion

for

the

driving

stroke

fram

,\

to

,\ butit

then

clears

the

film

for

the

return

skoke,

4

to

4

to

4.


3/49

5.3

Rotation

of

a

Vector

-

The

Rigid

Body

Rotation

Matrix

The

analytical

methods

of kinematic

svnthesis, to

be developed

in

the next

few

sections,

will

require

that we

have

some means

for expressing

the vector

that

results

from

the

rotation

of a

given vector about

a

specified

axis.

For

our

purposes,

the

vector to

be

rotated

represents

a link

in a

mechanism,

although

the methods

we

will

use

may

be

applied

to

any

vector. Since

we are

sfudving planar

mechanisms,

we

will

sfucly the

rotation

of

a

vector

in a

plane.

In

Chapter

9 these

ideas

will

we

generalized

to three

dimensional motion.

As shown

in Figure

5.7 a

vector r

in

the xy plane

has

components

r,=rCoSQ

(5.1)

and

r,

=

rsitlS

(5.2)

where

r is the

magnifude

of

the vector

r.

Note

that the

tail

of this

vector

passes

through

the

origin

of

the

xv coordinate

system.

Rotating

this vector

about

the

z

axis

by an angle

@

produces

a

new vector

r'.

The components

of r'

are

given

bv

r'= r'cos(0 +

0)i

+

r'sin(g

+

0)i

(5.3)

where

r' is the

magnifude

of the vector

r'.

The magnitudes

of

the vectors

r

and

r'are,

however,

equal

since

t'

was

produced by

a rotation of

r. Using

the fact

that

lr

I

=

lr'l

and the

trigonometric

identities

for

the

sine

and

cosine

of sums

of angles,

Equation

(5.3)

may

be

rewritten

as

1'=

dcosg

cos0

-

sing

sinO)i +

r(sing

cose + sino

cosg)

'

(5'4)

Recalling

the expressions

for

r,

and

r,

from

Equations

(5.1)

ancl

(5.2)

we

can

write the

above

expression

for

the vector

r'

in

terms of

the vector

r

as

r'=

(4

cose

-

I

sinO)i +

(4sine+ cose)j

(s.s)


4/49

Recognizing

that the components

of

r'are

linearly

related to the

components of r,

this

expression

can also be

written

in matrix

form

as

or

[r']= [n][r]

$.n

The

matrix

[R]

is

known

as

the

planar rigid

body

rotation

matrix.

It

is

an

operator

which

transforms

the

vector

r into

the vector

r'.

It

is

important

to recognize that

this

expression was

derived for

the rotaUon

of

a

vector passing

through

the

origin

of

a

coordinate

system,

and

that the

axis of rotation

also

passed

through

the

origin

of the

coordinate system.

The

matrix

[R]

has some

interesting

properties.

The inverse

of

this

matrix

is equal

to

its

transpose

[n]'=

[n]-'

(s.8)

While

this

can be

proved

rigorously,

it can also

be seen by

simply

multiplying

[R]

by

its

transpose and

observing that

the

result

is the

identity matrix.

It

can also be

shown

that the determinant

of

[R]

is

always

+1.

This is

known

as

a

proper transformation

since

it represents

an acceptable

rigid

body

motion

(ref).

In

a homework problem

it

will be

demonstrated that

a

determinant

of

-1

corresponds to

kinematically

impossible,

or

improper motion

of a

rigid body: furning

it

inside

out

EXAMPLE

5.1:

Find the vector

that

results

from

a rotation

of

the vector r

=

10i by'135,

180,

and

270 degrees

about

the

z axis.

Before

proceeding to the solution given below,

which

makes

use

of the rotation

matrix, you

should

first

solve this problem

geometricallv.

['',.l_

fcos0

-sinel[rl

L'',1

-

|

sine

cose

l[r,J

For

a 135 degree

rotation


5/49

For

a 180 clegree rotation

r'

r

=

[T;l,';;i,:li;;i][';]

=

ll

I)

,-r,',

-

fcos(180)

L r

I

sin(180)

h'r

=

l:ftTi

":l #i][';]

=

[

?,]

-sin(raoll[tol

=

[-1ol

cos(rsoilloj

L

o

I

For

a 270 degree

rotation

5.4

Kinematic Inversion

The concept

of kinematic

inversion

is

important for

the understanding of

graphical

techniques of kinematic

synthesis

as

well

as

many

other aspects of

kinematics.

Given

a

mechanism, its inversion

is

created

by

allowing

the

ground

link

to

be

mobile, and

letting

any

other

link

be

attached

to the

frame.

This is

illustrated in

Figures

5.8a

and

5.9a,

where the

mechanism in

Figure

5.8a

is inverted

in

Figure

5.9bby letting

the

coupler be the

ground

link.

A

sequence

of positions

for

the initial and inverted

mechanism

are

shown

in

Figures

5.8a-c

and 5.9a-c.

These

positions

were

chosen such

that the angle between

links

Qr{

and

QQ

were

the

same

in

both mechanisms.

It

is

important

to realize that

in

these two

mechanisms

aII

relahve motions are the same.

Thus,

@,

=

0,

,

@,

=

$r,

@,

=

0,

and Oo

=

$0.

The

equivalence

of relative

motions is

not

limited to angular positions.

Distances

between specified

points are

also

the

same

in

both mechanism.

For example,

AO,

=

A'Q,

BO,

=

B'Q

and

PQ

=

PO,

etc. The

equivalence of

these

and

other distances

in

both

mechanisms

can

be

verified

by

making

measurements on the figures.

5.5 General

Characteristics

of Mechanism

Synthesis

A

review

of the literafure would

show that

a

large number

of

methods have

been

developed

for

the

synthesis

of

four

bar mechanisms.

These

include both graphical


6/49

and analytical

methods.

Graphical methods

are

of interest

since

they

illustrate

some

of the basic

concepts of

mechanism

synthesis. Graphical

methods

for

svnthesizing

motion,

path anci

function

generator mechanisms

will

be

illustrated

beiow. Whil;

graphical

methods

have

been

generally replaced by

analytical

methods,

programmable

computer aided

drafting

svstems

have

made the graphical

approach

a

viable

alternative

to analvtical

methods.

Using

CAD

programmed

graphical

methods

it is sometimes

possible

to

see

how a

mechanism may

be

modified to

improve the

motion

that

it produces

(ref).

In addition

to the graphical

solutiory

one,

generally

applicable analytical

approach

will

also be

developed

for

motion, path

and

function

synthesis.

This

basic

approach

is based on a vectorization

of

methods

often

implemented

using

complex numbers to represent planar

vectors

(refs).

We will design

mechanisms to

provide

a

specified

movement.

However,

the

contirut.ous

desired

movement

is

not

specified

in

the

design. Rather,

selected

positions

are

specified,

and

the mechanism is

designed to

conform

to

these positions.

In

between

these

positions,

the

motion

produced by the

mechanism

may

not

fit

the

specified

motion.

5.5 Motion

Synthesis

We

will begin the

sfudy of specific methods of

four bar mechanism

synthesis

by

considering

motion synthesis

or rigid bodv

guidance.

In

this

case

we want

to proiuce

a specifiecl

rigid

body

motion

of an object

which

is attached

to the

coupler of

the

four

bar mechanism.

Three

positions

of a desired rigid body

motion

are

shown

in Figure

5.10.

Clearlv,

the

desired motion

is continuous,

and includes positions

between

those

shown,

but

the synthesis

will

be based on

just

these

three positions

known

as

accuracv

points

or

precision

points.

While

this may

appear to

be an

arbitrary

choice,

we

will

see

that

the

number

of accuracy

points

has

a

profound influence on the

solution procedure

used

to

size

the

links

in

our

design. We

will

develop both

graphical

and analytical

methods

for

synthesizing

four

bar

mechanisms.

5.6a Motion

Synthesis

- Graphical Method

In all

of

the

graphical

solutions, subscripts

1,2

and

3 will

be used to

denote

the

location

of a

point

in

the

firsf

second

and

third

accuracy

point.

If

a

subscript

is

not

assigned

to a

point

then

it

refers to the

first

accuracy

position, that

is,

A

has

the

same

meaning

as

4.

We

will

start

the

graphical

construction

by assigning

locations

of the

moving pivots


7/49

A and

B

on the coupler link

as

shown

in

Figure

5.11.

This choice

of

locations

for

A

and

B

was

arbitrarv.

The ground

pivots,

Q

and

Q

are

the

centers

of the circles haced

by points

A

and

B

respectively.

To

find these

centers we

construct the

perpendicular

bisectors of the lines

ArA,

and

ArAr.

The ground

pivot

Q

is located

at

the intersection

of

these

two

bisectors:

a

point

equidistant from

4,

4

and

4.

Similarly,

the

ground

pivot

Q

is located

at

the intersection

of the

perpendicular bisectors

of

B,B,

and

BrB..

We have

now

svnthesized

a

four

bar mechanism,

qABq,

that

will

transport its

coupler

through

the three specified positions.

This graphical synthesis procedure

is

now

summarized

in

the following

steps:

1.

Pick three positions

in

the

rigid

body motion

that

r,r'ill

be

used

as

accuracy

positions.

2.

Assign locations to

the moving

pivots

A

and

B

as in Figure 5.10.

3.

Construct the perpendicular

bisector

of ArA, and

At,\.

The intersection

of

these

bisectors

determines the location of

the

fixed

pivot

Q.

4. Construct the perpendicular bisector

of

BrB,

and

BrBr. The intersection of

these

bisectors determines the location of the

fixed

pivot

Q.

5.

Compare the

mechanism characteristics

with design specifications

such

as the

overall

rigid

body

motion

and

the transmission

angle. Redesign

the

mechanism

as

needed bv modifiiing

the free

choices.

However,

there

is no guarantee that this mechanism

will

suit

all

of

the designers

specifications. The mechanism

only

reproduces the

specified

accuracy

positions.

The

motion

of

the

coupler link,

between

the specified

positions,

may not

be

acceptable.

It

is

also

possible

that the

ground

pivots

may fall in

unacceptable

locations

on the

frame,

the

input

and

output

links

may

sweep over

an

unacceptable

region

of the

frame, the

transmission

angle

mav not

be acceptable

at

some

mechanism

positions

or that

the

overall

mechanism

mobility

may not

be

compatible

with

the designers

need.

For

example,

the designer

may need

a crank

and rocker mechanism,

but the

svnthesis

procedure

may

have

vielded

a drag link mechanism.

While

there

mav

be

many

potential

problems with

the mechanjsm we have svnthesized,

there

are an infinite

number

of mechanism

that

can be found

to produce

the specified

motion

when

three

positions

are

given.

We

can

begin to

identifu

other mechanisms by recalling

that

the locations

of the

moving

pivots,

A and

B,

rvere

arbitrary.

Placing

one

or both of

these

moving pivots

at

a

different location

on the coupler

will

produce a

different mechanism.

It

is

important

to recognize

that

the

moving

pivots may be

placed

anywhere

in

space

as long

as they


8/49

are atbached

to the part whose

motion we

are

tr"ving to

produce.

Another

mechanism

synthesized to produce

the

motion

shown

in

Figure

5.11 is

shown

in

Figure

5.12.

Note

that

in

Figure

5.12

extensions were

added

to the

moving

link

to

provide

locations

for

the

moving

pivots,

but

the

motion

of the

moving

link was

not changed.

If

this second

mechanism

is

still not

acceptable

we

can

relocate

the

moving

pivots

and svnthesize

yet

another

mechanism.

In searching

for different

mechanisms, the

designer

is

making

use

of

"free

choices",

that is the freedom to position the moving

pivots to produce

an

acceptable

mechanism.

The fact

that

we have

these

free

choices,

or

free variables,

at

our

disposal

in

the synthesis

process,

implies that

there

are an

infinite

number

of

mechanisms

that

can

be

designed to

produce

a

prescribed

motion,

at least

when

the

design is

based

on three

accuracv

points.

5.6b

Motion

Synthesis -

Analytical

Methods

Motion

synthesis

is concerned

with

the design

of

a

mechanism

to produce

a

specifiecl

rigicl body

motion of the coupler

link.

As

such, the clisplacement

of

a

point

on

the

coupler,

and the

rotation

of the coupler

are

specified. The

analytical approach

to

motion

synthesis is

based

on

a

vector

loop equation around

the mechanism

in

groups of

two

of its accuracy

points

as shown

in

Figure 5.13.

Writing the

vector

loop

equation

from

Q

to

Q

over mechanism in

position

one

and back

to

Q

over the

mechanism

in

the

k-th

position

vielcls:

p+q-

s-r+rr

+sr

-Qr

-P,

=

0

From

Figure

5.13

we

can see that

P+q-9r

-Pr

-

-dr

and

-f-S+Sr+f*=d*

(s.e)

where

d*

is the displacement

of

a

point

on the coupler.

Comparing

Equations

(5.9), (5.10)

and

(5.11)

we see that

the loop equation

may

be

satisfied

by satisfying

Equations

(5.10)

and

(5.11)

individually.

In

terms of

the

synthesis,

this

means

that

we

can

find

the vectors

p

and

q

independent of

the vectors

r

and s. This

mav not be too

surprising

given our experience

with

the

graphical

approach

where

we determined the

input

and

ouQut links

based

only

on

their

respective

moving pivot

locations.

That is, the

position

of

the

moving

pivot

A,

determined

Q,but

had no effect on the

position

of

q.

(s.10)

(s.11)


9/49

We

will develop

the solution of the synthesis equations

for

the vectors

p

and

q

from

Equation

(5.10).

The solution for

r

and

s from

Equation

(5.11)

will

be exactly

the

same

as that

for

p

and q. The

vectors

p*

and

q*

in

Equation

(5.10)

are related

to

p and

q

by

a

simple

rotation

lp*l=

lP-l[p]

[q*]= iq-liql

(5.12)

and

where

[4]

and

iQ*]

u..

the rotation

matrices defined

in Equation

(5.6).

Substituting

Equations

(5.12)

and

(5.13)

into

Equation

(5.10)

leads

to

lP-

-

tl[p]+

[e*

-

r]lql

=

lo-l

(5.13)

(s.14)

where

[t]

is tne

identity

mahix.

This is

an equation

in

the

unknown

link

vecto.s

[p]

and

[q]

and the

rotation matrices

[4]

anct

[q*].

fnis

is

the

fundamental

equation

of

kinematic

synthesis.

We

will

see

that the equations

for

path

and

function

synthesis

are

of

the same

form.

Before

beginning

to develop the solution

procedure, it

must

be

recognized

that

Equation

(5.14)

relates the

frst

and k-th accuracy

points

of the

mechanism.

If we

are

synthesizing

a

mechanism

by specifying

three

positions

then we

have two

equations

of the

form of

(5.14),

one which

relates the

frst

and second

positions,

and

one which

relates

the

first

and

third

positions.

We

will

concentrate

on

the solution

for

the

three

position

case.

Our

goal, now, is to develop

an

approach

for

solving

this

equation.

Using

Equation

(5.6)

we

can

explicitly

write

the rotation

matrices

ffr]

ana

ffr]


10/49

lRl

=

fcosO,

-sinO,-l

L

zr

I

sin@, cosO,

]

lcoso^

-sino^l

LP,l

=

[rino,

coso;J

where

the angles

@,

and

@,

are

the rotations

of

the

input link,

p,

from the

first

acflrracy

point

to the

second,

and

from

the

first

accuracy

point

to the third.

These

angles are unknown.

Similarly,

the

matrices

[Qr]

u"a

[Q,]

ur"

where the angles

S,

and

0,

are the coupler

rotations

when

moving

from

the

first to the

second

mechanism

positions and

from

the

first

to

the

third

positions.

These angles

are

known:

they

are

specified

by

the designer

since

this

is

motion

synthesis.

We

can

now

summarize the equations and the

known

and

unknown

quantities

in

these

equations.

For three

position synthesis we have

two planm vector

equations

of

the

form

of

(5.1a).

One

for

the displacement of the mechanism

from

accuracy

point

one

to

accuracy

point two

lP,

-

tl[p]+

[e,

-

t]lql

=

lo,l

(s.17a)

and one

for

the

displacement of the mechanism

from

accuracy

point

one

to accuracy

point

three

[P,

-

r][p]

+

fe,

-

tl[q]

=

lo,l

(5.18b)

These

two vector equations

represent

four scalar

equations.

The

known

quantities

are

re,r=[:TJ;

":iJ;]

re,r=

[::;f;

":i,l,]

l0


11/49

the displacement

vectors

d,

and

d,

of

a

specified point

on the coupler

and

the

rotations

$,

and

$,

of the

coupler.

The

unknown

quantities

are

the vectors

p

and

q

and

the

rotations

O,

and O,

of the

input

link.

These are

six

scalar

unknowns

which

must satisfy the

four

scalar

equations

resulting from

(5.17a)

and

(5.17b).

This

seemingly

awkward

sifuation,

more unknowns

than equations,

has a

simple

resolution: we arbitrarily

assign

values to

two

scalar

unknowns.

Again, this sifuation

should not be

surprising

based

on

our experience

with

the

graphical solution

to

motion

synthesis. We saw that

the designer

was

free

to

position each

moving pivot

on

the coupler,

which

resulted

in

an infinite

number

of possible mechanisms.

In

the

analytical solution we

also see

that

there is

an infinite number

of solutions: this is the

implication

of having

more

unknowns

than equations.

However,

as

we

will

now

see,

the nafure

of the

analytical

solution

will

depend

on

which

variables

we

assign.

If

we assign anv

two scalar

components

of the

vectors

p

and

e,

for example

pr,

and

4r,,

then

the variables

p*

Q,r,

@,

and

O,

will

be

unknown.

Equations

(5.17a)

and

(5.17b)

can

then

be

solved

for these

variables.

However,

these

equations are nonlinear

in

the angles

@,

and

@r,

and

a

nonlinear, numerical, solution

algorithm

such

as

the

Newton-Raphson method

would

have

to

be

used. Alternatively,

we could assign

values

to

O,

and

@r,

in

which

case the

four scalar

components of the

vectors

fp]

and

fl]

wouta

be unknown.

In

this case,

Equations

(5.77a)

and

(5.17b)

would be

linear

in

the

unknown

quantities and

are

given

in

matrix

form

by

lQ,

-

rrlItpr.l=

[td,]l

tq,

-

rllltqll

LIo,tl

Ilq

Lre

-rl

-

rI

(5.18a)

or

in

"shorthand"

form

lslln]=

lpl

(5.18b)

where

each

of the

partitions in

the

coefficient

matrix

represents a

2 x2 arcay ancl

the

partitions

in

[n]

and

fn]

are2x1

arrays.

Using

Equations

(5.15)

and

(5.16)

the

coefficient mahix

in

Equation

(5.18a)

can

be written

in

a

more

explicit form

as

l1


12/49


13/49

l-o.uo

-o.833

-o.658

+.q+ol

tst

=

I

0.833 4.446

o.s4o

,.urr

I

L

r

I

-1.904

4.427

-1.174

-0.985

I

lo.or,

-r.eo4

0.e85

-t.t

o)

I

tr+r

0.208

-0.785 -o.ttzf

lsl_,

=

|

-0.208

t.t4t 0.372

-0.785

I

L

r

l

-1

413

0.433 0.606 0.38r

I

l-0.+lo

-t.4ti

-0.38r

o 606

j

l-o.ztz

-0.685

-o.658

-o.e+ol

fst

=

I

0.685

4.272

o.e4o

+.osa

I

L

r

|

-1.534

{.8,16

-1.174

-0.985

|

I

o.t+o

-1.s34

o.eB5

-t.n+

l

and the

inverse

of

[S]

is

Using Equation

(5.19),

p

and

q

ane:

p

=

5.041-

1.085J

q

=

0.200I

+0.400J

Which

are similar to the

graphical solution

shown

in

Figure 5.10.

For the

dyad

composed

of the

output lir:dr'

qB

and coupler segment

8P

the

displacement

vectors

d,

and

d,

and the

coupler rotations

$,

and

S,

are

unchanged. Picking

the free

choices

as

ltz

=

43.24'

and

Vt

=122.25"

and

using

these angles

in

Equation

(5.18c)

leads to

and

its inverse

l3


14/49

o 062l

1.0s3

I

o

118

I

0667l

I t.ttt

rrl=

|

:,^::,,

[-o

rn,

-0.444

1.337

0.793

-1.401

-1.053

4.062

0.667

-0.118

From Equation

(5.19)

the vectors

describing

the

right

hand dyad of

the mechanism

are

found

r=5.99I+1.78J

s

=

0.199I

-1.2031

By

now

you

may be

wondering

why

we have concentrated on three

position

synthesis? The answer lies

in

the nafure of the

solution procedure

for the synthesis

equations.

Specifving more than three mechanism

positions

leads

to

an inherently

nonlinear

set

of

equations. That

is,

we

cannot

assign

values to

some

of

the

unknown

variables and produce

a set

of

equations

which

are

linear

in

the remaining

unknowns,

if more

than

three

positions

are

specified.

This

point

is

illustrated

in

one

of the

homework problems.

5.7 Path Synthesis

In

path

synthesis,

only

the curve traced bv

a

point on the coupler

if

of

interest,

there are no constraints on the angular

position

of the

coupler as we had

in

motion

synthesis. We

will

illustrate

the

synthesis procedure using three accuracy

points

on

the

desired continuous

path.

The

graphical procedure

depends on the

number

of

accuracy points.

The

fornutlntiott

of the analytical

approach

is independent of

the

number

of

accuracy

points, however,

the

solrttioll of the synthesis

equations

will

depend on the number of accuracv points,

and

how

free

choices are chosen

as

in

motion synthesis.

5.7aPath

Synthesis

-

Graphical Method

A

desired

path,

and three positions

on

that

path

which

will be

used

in

a

mechanism synthesis

are

shown

in

Figure

5.14a.

It

is

important to

realize that

the

positions

Pr,

P,

and.(

represent three locations of

a

point,

P,

fixed

on the coupler.

The

point P

moves

through

the three positions

\,

P, and

{

as

the

mechanism is being

driven.

Realizing that

P

is the location

of a

point

fixed to the

coupler will help

our

understanding of

the

graphical synthesis of this

mechanism.

The graphical

approach

will

make

use

of the concept

of

kinematic inversion.

It

should

be

remembered

that

an

l4


15/49

inverted

mechanism

maintains

the

same

relative motion

as

the

original

mechanism.

To synthesize

a

path

generating mechanism

we

first pick,

arbitrarily,

the length

and orientation

of the

ground

link,

and the length

of

the

input link.

One selection

of

these

links is

shown

in

Figure 5.1.4a

along

with

the

path

we

are

trving

to svnthesize.

The

window

in this

figure

also

shows

a

generic

four

bar

mechanism

to

be

used

as

a

reference

for identifving

the labels of pivots

and

visualizing

the relative

motion of

links.

Refurning

to the synthesis,

we arbitrarilv

pick

the length

of

one

side of the

coupler from

A

to

P.

This allows

us

to

determine

three

positions of

the

moving

pivot

A,

4, \

and

A,

are each

locatecl at a distance AP

from

\,

P,

and

{

respectivelv

(Figure

5.14b).

There

are two

possible intersections that determine the locations

of

these

points.

We picked the intersection

so that they

followed in sequence

from

,4,

to

4

to,4, as

the

input link

rotates

counterclockwise. If we

coulcl

locate

the

moving

pivot,

B,

then the mechanism

woulcl

be

complete.

To

locate B

we will

invert the

mechanism

at

the

first

accuracv

point,

making AP

the

fixed

link,

and

QQ

"

moving

link.

In

this

inverted

mechanism

B,

which

is fixed,

is

the center of the circle traced

by

Q.

W.

will

now

locate

the

positions

of

links

QA

and

QQ

in

the

inverted

mechanism

at the

second

and

third

accuracy

points.

The points

Q

ancl

O,

at

the

second

and

third

accuracy points,

will

be

denoted by

(O,)r, (q)r,

(Q),

and

(q)r.

At

the

first

accuracv

point,

Q

ancl

Q

will

be

named

without the

additional

subscript.

In the true mechanism

the angle

y,

between

the input

and coupler

links

decreased

when

going

from

the

first

to

the

second

accuracy

point

(Figure

5.14c). In the

inverted

mechanism the

same relative

motion

must be

maintained.

The

position of

Q

in

the

second

position

of the inverted mechanism,

(Q)r,

is

located

as

shown

in

Figure

5.14d.

The

point

(Q

),

must

lie

on a circle

with center at

(Q),

and with

a

radius

of

QO.

as

shown

in

Figure

5.14d,

and labeled circle

#1. To locate

(Q),

ot

circle #1 we

use the

fact that the distance

between

P and

Q,

ut

measured

in

the true mechanism,

must

be

the

same in

the inverted

mechanism.

Recognizing

that

at

the

second

accuracy

point

P

is at Pr, we

now draw

the circle

with center

at

\

and

radius

Prq.

Where this

intersects circle

#-1, determines the location of

(Q)r.

But there

are

two

such

intersections.

Can we choose either of

them?

In

general

only one

of

these

intersections

will

yield

(Q),

in

a

position which

is

consistent

with

the inversion of

the

mechanism.

The

intersection

marked

in

Figure

5.14d

is

consistent

with

the

inversion

as can

be

seen bv

comparing the

angle between

links

QA

and

QQ

in

the

true

mechanism at the second accuracy

point

(Figure

5.14c) with that

in

the inverted

mechanism. We

could

also

identif"v the correct intersection by comparing

the

l5


16/49

diagonal

44

i"

the

true

mechanism

with the

corresponding diagonal

in

the inverted

mechanism.

These

distances must

be

the

same

in

both

mechanisms.

The positions

of

Q

and

Q

at

the

third

accuracv point

are

found in

exactly the

same manner as

were

(Q

),

and

(q)r.

These are shown

in

Figure

5.14e. We now

have three positions

of

Q

in

the inverted

mechanism. These

points

q,

(q),

and

(Q

),

lie

on

a

circle

with

its

center

at B.

The

intersection of

the perpendicular

bisectors

to

qQ),

and

(q)r(q),

locates

point

B

(Figure

5.141). The mechanism

qAPBq

is our path generator shown

in

the

first

precision

point in

Figure

5.149. This

synthesis procedure is summarized

in

the

steps

listed below.

1.

Pick three

accuracy

points on

the continuous

path.

2.

Assign

the

length and

orientation

of the ground

link

(Figure

5.14a)

3.

Assign the length of the

input

link

(Figure

5.14a)

4.

Assign

a

length to

the segment of the

coupler

from A

to

P

and then

graphically

determine the

location of the points

4,

4

and

.\

(Figure

5.14b).

5.

Draw

the input lir:Jr-

qA,

and the coupler

segment

AP at

the

three accuracy

points

(Figure

5.14c).

Determine

the angle

y

at each

accuracy point.

6.

Invert

the

mechanism,

making the

coupler

the

fixed

link. Graphicallv

locate

the

position of

Q

in

the

second

accuracy

point

(Q

),

(Figure

5.14d).

Now, locate

the

position

of

(Or),

at

the intersection

of

two

circles, one

with

radius

QQ

and

center

at

(Q

),

and the other with

radius

Prq

and

center

at

{

(Figure

5.14d).

Remember,

that

while

there

are

two

intersections

of

these

circles, only

one

represents

the

inverted

mechanism.

6.

Locate

(Q

),

and

(4

),

i.

the

third

accuracy

point in

the

same

way that

these

points were

determined

in the

second

accuracy

point

(Figure

5.14e)

7.

Construct the perpendicular

bisectors

of

q(q),

and

(Q

)r(q)r.

The intersection

of these bisectors

determines

the

location

of the

point

B.

8.

We

now

have

a

path synthesis

mechanism

in

the

first accuracy point

(Figure

sJ,

g). The performance

of this mechanism

should

be

compared with the design

specifications, and redesigned

as needed.

t6


17/49

As was observed

with

motion

synthesis,

it

is possible that the

first

mechanism

that

we

synthesize

will not

meet all

of our

design

criteria. These

may include

size

of the

mechanism, position of the

ground

pivots, and

transmission

angle,

among others.

If

this mechanism

is found

to

be

unacceptable

then

any

or all of

the free

choices

may be

changed, and

another mechanism

synthesized.

The free choices

for

this

synthesis

include

the length and orientation

of the

ground

link, the length of

the

input link and

the length of the link AP.

5.7b

Path Synthesis

-

Analytical

Method

The analytical approach to

path

svnthesis follows

that developed

for motion

synthesis. For a desired path,

displacement

vectors

d,

anddr,

between accuracy

points,

are specified. These

are

defined in

the

same way

as the coupler displacements

in motion synthesis.

Vector

loop

Equations

(5.10)

and

(5.11),

developed

for motion

synthesis,

are

then

applicable

to path

synthesis.

For three position path synthesis we

can then use

Equation

(5.17)

to synthesize

one

dyad

(the

input

link

and

one

part

of the

coupler) and

an

analogous

set

of equations

to

synthesize

the other

dyad

(the

output

link

and the remaining half of

the

coupler).

The equations

for

the

input -

coupler

dyad

are

[e

-

t][p]+

le,

-

tl[q]

=

[0,]

[q

-

t]lpl *[e,

-

t][r]

=

[0,]

(5.20a)

(s.20b)

which

are

the

same

as

(5.17a)

and

(5.17b).

We

again have

two

vector or

four scalar

equations,

but now

there

are

eight

scalar

unknowns, pr,,

pry,

e1,,

4r,

@r, @r,

0r,

0,

and assigning

values

to @,

,

@r,

6,

and

$,

results in

a

set

of linear algebraic equations.

By assigning

values

to

the input link

rotations,

@,

and

O,

we

are

prescribing

the

position

of this

link with

respect

to

the

position

of

the

tracing

point

P on the coupler.

This is

known

as

path synthesis

with prescribed

timing.

EXAMPLE

5.3

We

will

design

a

mechanism

to synthesize

the path shown

in

Figure

5.1.4a.

We

will

determine

our

free

choices

from

the

graphical solution

which

should

result in

a

mechanism

that

is

similar

to that

shown

in

Figure

5.14g.

Since

this is path

synthesis,

only

the

displacement of

a

point

on

the

coupler

is specified

in

the design.

For

the path shown

in

Figure 5.-1,4a,

the displacement

of point

P

is given by the vectors

l7


18/49

d,

=

-0'

15I

-

0.38J

dl

=

-1'391+0.241

For the

dyad composed

of the input lir*,

qA,

and the

segment AP

of the coupler,

the

free

choices

are

selected as

@,

=

21.43' and

@,

=

73.3T

for the

input

link

Q,4

and

d,

=

-7.05'

and

S,

= 5.81'

for

the

coupler

link.

The

coefficient

matrix

[S]

from

Equation

(5.18c)

is

-0.365

-0.008

-0.069

-0.123

-0.958

-0.005

-0.713

0.101

Taking the

inverse

of

[S],

and using Equation

(5.19),

the vectors

p

and

q

found,

P

=

0.

4891+0.8671

=

O,A

I=3.95I+1.88J=AP

These

results

are similar

to

those

found

graphically in

Figure 5.1,4g.

For the dyad composed

of the

output

lirk

qB

and the

coupler segment

BP,

the

displacement vectors

d,

and

d,

are,

of

course,

the

same

as given above.

Likewise,

the

chosen

values

for the

coupler rotations

S,

and

$,

must

be

the

same.

The

output

link

rotations

are

chosen ars,

lrz

=

-1A

and

ry,

=

-137.5".

Computing the

matrix

[S],

and

using Equation

(5.19)

we find

r

=

0.4761-

0.2891

s

=

0.776I+3.591

5.8 Function

Synthesis

Concepfually,

function

synthesis is

quite

different from either motion

or path

[-0. ooq

tst=|

0365

L

J

I

-0.713

Iorrt

o.tn

1

-0msI

-0

101 I

-o

oo5l

l8


19/49

synthesis.

In function

synthesis

we are

concerned

with

producing

a coordinated

motion

between

the

input

and

output links.

We

are

not

explicitly

concerned

with

any

motion

of the

coupler. Despite

these

concepfual

differences

we

will see

that the

form

of the equations

used

for function

svnthesis closely

follows

that

used

for motion and

path

synthesis.

The overall

goal of

function

sl.nthesis is to produce

a

mechanism

that

generates

the

function

y

=

f{x)

as

shown

in

Figure 5.15. When

the

input link

"points"

to

a

value

of

the inclependent

variable,

r,

the

output

link

"points"

to the

corresponding value

f(x).

Any

mechanism

that

we

design

for

this purpose will

onlv

produce

the

functional

relationship

accurately

at

the

precision

points as shown

in

Figure 5.16.

The difference

between

the

synthesized

and

desired

function

is

known

as the

strucfural error.

Minimization of the

strucfural error

would

appear to

be a

desirable feafure

of any

function

svnthesis

technique.

We

can

think

of the svnthesized

function

as a

polynomial

approximation

of the

specified

function.

It can be

shown that

a

Chebyshev

polvnomial

deviates the

least

from

a specified

function.

To obtain

the

best fit

of

a

Chebyshev

polvnomial

to

a set

of

data

we pick

values

of

the

independent variable,

x,

according

to

the

formula

(s.12)

whereT is

an index

corresponding to the number

of

the precision

point,

Ax is the

range of

r

over

which

the

function

is to

be

synthesized and

n

is the

total number

of

precision

points.

Unlike motion

or

path

synthesis, the

location of precision

points

in

a

function

generator

are

not

arbitrarily assigned

by

the

designer.

However,

the

designer

does specify the range of

r

for

which

the mechanism

will

be

synthesized,

as

well as

the range

of rotation of the

input

and

output links

that corresponds

to the

range of

variation of

the

independent

and dependent

variables. In

fact,

an important

relationship exists

between these

variables and the

angular

position

of the

input

and

output

links.

The range

of rotation

of the

input

(output)

link

is

scaled

by the

range

of

the independent (clependent)

variable, and this

scaling is the

same

for intermediate

rotations of the

input

(output)

link

between

precision points

xi=xo-+['-*'(1;)^]

t9


20/49


21/49

xi

v;

0

0.0 0.0

1

0.134

0.018

2 1.000 1.000

a

J

1..866 3.482

4

2.000

4.000

We

are

now in

a

position to calculate the relative

rotations

of the

input and

output

links

from

Equation

(5.22)

o,

=+

(x,-x,)

'

Ax'

o,

=

9(t.o-

0.134)

=

25.98

o.

=

+(1.866-0.134)

=

51.96

Aw

V,=

^LOr-Yr)

^y

vr,

=ry(r.ooo

-

o.or 8)

=

22.10

4'

90_

Vr

=

+

(3.482

-0.018)

=77.95

The calculated

rotations of

the

input

and

output links

are shown

in

Figure

5.17.

5.8a

Function

Synthesis

-

Graphical Method

The graphical

approach

will now be introduced

using

the results of Example

5.4.

First,

we arbitrarily

fix

the

length

and orientation

of

the ground and

output

links

as

shown

in

Figure

5.18a.

If

we could locate

point A,

consistent

with

the desired

function,

we

would

have a complete mechanism.

ff

we

invert

this mechanism,

making the

input

link the

fixed

link, then the

unknown point A

is

at

the

center of

a

circle traced by

point

B. We

can

perform

this inversion

since

we have computed

the

relative rotations

of the input

and

output links

Or,

03,

V,

and ry'

In the true mechanism the angle between the

input

and

ground

links

increased

by

25.98

deg.

when

moving from

the

first

to the

second

precision point.

This same relative motion must

be

maintained

in

the inverted

mechanism, therefore

QQ

mustrotate

clockwise

by

25.98

deg.

This

locates

the

second

position

of

O, which we are

calling

(Q

),

Gigure

5.18b).

To locate B,

we

2t


22/49

observe

that

in

the true mechanism

QB

rotates counter

clockwise.

We therefore rotate

(Or)rB

counterclockwise, by

22.10

deg,

relative to

(Q)rB'.

This

establishes

the

position

of

B,

(Figure

5.18b).

The

positions

of

(Q),

and

B,

are

determined

in

the

same

manner

as

(q

),

and B,

as

shown

in

Figure 5.18b. We

now

have

established

three

positions

of

B

in

the

inverted mechanism,

B,

B,

and

Br.

These

points move

on a

circular path

with

its center

at

the unknown pivot

A.

The intersection of the

perpendicular

bisectors

to

BB,

and

BrB,

establishes

the location

of A

as

shown

in

Figure 5.18c. The

mechanism

qABq

in

Figure 5.18d is our desired

function generator

at

the

first

precision

point.

The

steps in

this graphical

synthesis

are

summ arized

below.

1.

Determine

the values of the independent

variable

at

the precision

points

using the Chebyshev

spacing

formula in

Equation

(5.21).

Determine the

scale

factors,

and the relative

rotation of the

input

and

output

links

from

Equations

(5.22)

2.

Assign

the

length

and

orientation of

the

ground

and

output links

(Figure

5.18a)

3. Invert

the

mechanism about the

(unknown)

input

link.

Graphically locate

the

positions

(Q),

and

Brby

rotating

the

ground

link by an angle

-O,

and the

output

link

by

and

angle

-y,

using the reference lines shown in Figure

5.18b.

4. Repeat

the

procedures

in step 3

to determine

the

locations

of

(q),

and

Br.

5.

Construct the

perpendicular

bisectors

of

BB,

and

BrBr.

The intersection

of

these

bisectors determines the

location

of the

pivot

A.

6. Evaluate the performance of the mechanism

with

regard

to

the design

specifications, and

redesign by altering

the free

choices as needed.

As

with our previous

designs,

we

made

a series of

arbitrary

choices of link

lengths

and orientations when we designed this mechanism.

We

fixed

the

length and

orientation

of

the

ground

and

output

links.

If

this

mechanism is

unacceptable, these

free choices can be

altered

and a new mechanism

designed.

An

interesting feafure

of

function

generators

is that we

can

scale all

link lengths by

the

same

factor

and still

produce the

same

function. Thus,

if

the mechanism

we design is to small or to

large,

it

may

be

simply

scaled

up

or

down.

The fact that

we

can

multiply

all

link

lengths

by

a constanf without

altering the

functional

relationship

between input and

output

link

rotations,

will be apparent

in

the

next section

on analvtical design of

function

generators.

22


23/49

5.8b

Funcfion

Generation

-

Analytical

Method

The analytical

procedure

for

dimensional

synthesis of

function generator

mechanisms

is

based

on

a vector loop equation around

the

mechanism

in two

positions

as we

did

in motion and path synthesis.

Flowever

in

function

synthesis

we

are

not

explicitly

interested

in

the

motion

of any

points

on the

coupler

link.

Therefore,

we

do

not need

to include

a

vector

to

a

tracing

point

P

on

the coupler.

For

a

mechanism

in

its

first

and

k-th

positiory

the vector

loop equation

is

given bv

p+t-r+rr

-t* -P*

=0

(5.23)

where

all

vector

quantities

are

defined

in

Figure 5.19.

Using rotation

matrices

to

write

r.,

t*

and p*

in

terms of

9

t

and p,

Equation

(5.23)

may

be written

as

lp--lj[p]*lr--l][t]=

rr-t=ln--l]lr]

(s.24)

where

the

right

hand

side was

written

in two

forms

to

display two important

physical

feafures.

First,

by examining

the

second

term

on

the

right

hand

side,

we

can

see

that

Equation

(5.24)

is linear

in

the

link lengths.

That

is, if

we

were to

multiply

this

equation

by

a constant

we would

change

the

link

lengths,

but we

would

not change

the

relationship

expressed

by

this

equation.

Secondly,

from Figure

5.19

it can

be seen

that

r,

-

r

represents

the

displacement of

point

B when the

mechanism

moves between

the

first

and

k-th

precision

point.

Recognizing that

the

right

hand side

of Equation

(5.24)

represents

the displacement

of

point B it

can be

written

as

lp-

-

rl[p]*

lr-

-

llltl

=

io-l

(s.2s)

where

it must

be

remembered

that d* is

the displacement

between

the

first

and

k-th

precision

points.

Equation

(5.25)

looks

similar

to

(5.14)

which

we used

for motion

and

path

synthesis.

However,

it

is

important to

realize

that

Equation

(5.25)

is

for

the

whole mechanism

as where

in

motion

and

path

synthesis

we had

equations

of

this

form

for

each

dyad.

For three position

synthesis,

we have

two

vector equations

of

the

form

of Equation

(s.2s)

le

-

tllpl

*

lr,

-

rlltl

=

fo,l

lP,

-

tl[p]*[r,

-

t][t]=

fo,l

(5.26a)

(5.26b)

23


24/49

The solution

procedure

for

Equations

(5.26)

is

similar

to that

used

for

three position

motion

and

path

synthesis.

We

will use

the fact

thaf

in

a

function

generator,

the link

lengths

can be scaled

by

a

constant

and assign the

length

of

the

output link. We

will

also assign

the

orientation

of this link

in

the

first

precision poinl

With these choices

the vectors

d,

and

d,

are

known

since we

have

previously

calculated the

angles

ry, and

V3.

With

these

assignments

we

now

have

four scalar

equations to find the six

scalar unknowns

p1,,

p1y,t1,,

trr,

S,

and

Sr.

Using

Q,

and

0,

as our free

choices leaves

us

with

four scalar

equations to

find four scalar

unknowns, the components of the

vectors

p

and

t.

24


25/49

EXAMPLE

5.5:

The

function synthesis

mechanism,

designed

above

by

graphical

methods

will now

de

designed

using

analytical

methods.

To

define the

vectors

d,

and

d,

we

must

first

assign

the position

of the

output

link,

at

the

first

precision

point,

with

respect

to

the

ground

link.

For

the example

the

initial position

of

the

output

link

was

chosen

to

be

he same

as

that determined

from

the

graphical

solution:

With

this

assignment

and

setting

the

length

of

QB

=

2.82

in.,the

vectors

representing

the

output

link

in

its

three

precision

points,

r',

r"

and

r"'

are

r'

=

2.82fcos(68.

28)I

+

sin(68.28)Jl

r"

=

2.82lcos(90.

38)I +

sin(90.

38)Jl

r

"'

=

2.82fcos(146.22)I

+

sin(l

46.22)Il

The

displacements,

d,

and

d.,

of

point

B are now

found

from:

d,

=

d"-d'=

-1.06I

+

0.20J

d,

=

d"'-d"=

-3.38I

-

1.05J

Since

the

relative

rotations

of

the

input

link,

@,

and

@,

are knowry

the

matrices

ffr]and

ffr]

are

easily

found.

Choosing

the coupler

rotations

to

be

6,

=

10

and

6,

=

-66,

the coefficient

matrix in

Equation (5.26b)

is

[-0.

ror

r,r=liffi

Iorrt

-0.438

-0.101

-0.788

-0.384

-0.060

4.342

-0.593

4.914

0s421

-0.060

I

0e14 I

-o

5e3J

25


26/49


27/49

'56

'58

82

80

48

46

Fro^

U,

s.

? ale"^|"

4,za+,211

Ftaulze

s.

t

*5o

[xreur

LruK

+L

S

2-


28/49


29/49

tr\Gu

Oe

\ vt

qoln

vnec.L,o:nis

r,rr

\i,ne

ovat

o,t

teast

?*t

of

,#

,,+,

s.4

pgr,vl

?

qeuetalesa

t+.oCt"t

lils

Tofu"


30/49

FI6UBE

s.

Q2

ta

tr

l6u

rL

s.?


31/49

ol

(a)

A

F(qu?e

s.6

trGu?

s.1


32/49

?osit'o^

3

?osit

o.,

Z

?osit'',n

\

F\quRE

S

to


33/49


34/49


35/49

2t'

=

3?J0ZlJ


36/49

Tesired

)

I

a*h'

Circl

Documents

4 Bar Mechanisms