UNIVERSITY

OF NEWCASTLE

UPON TYNE

AND CONTROL

DYNAMICS

of

-

LINKAGE

MECHANISMS having

TWO DEGREES OF FREEDOM

BY

A. H.

A Thesis Degree

of

Faculty University

YOUSSEF

M. Sc.

submitted

for

Doctor

of

Philosophy

of

Newcastle

r

March

in

Science,

Applied

of

the

1974

upon

Tyne

the

BEST COPY AVAILABLE

Variable print quality

SYN0PSIS

the

The work

described

dynamics

and

having

two

basically,

their

the

second

derivation

of

solution studies

concerns

linkage

planar controlling

experimental the

to

more

analysis.

I

generate the

closely.

investigations

numerical

mechanisms The work,

solutions

The first

topics. equations

of

motion,

linearised

and of

concerns

linkage

resonance of

a given linkage

a general output

inputs

The third to

analysis

and stability.

optimisation

some of

an output

of

three

the

thesis

freedom.

of with

numerical

involving The

degrees

this of

control

deals

concerns

in

check

and the

and

to

generate

concerns the

also

validity

linearised

ACKNOWLEDGEMENTS

I

I

for

Maunder

his

Mr.

Hewit,

to

C.

My thanks

Engineering

to of

staff

office

for

the

experimental

the

figures

in

to

this Lastly,

to

my wife

understanding.

the

Oldham

K. helpful

workshop

Department help

their

this

for

also

Dr.

Burdess,

their

the

the

and of

Mr.

and

in

J. R. and

comments.

and design of

Mechanical

manufacturing

and preparing

rigs

My thanks typing

J.

for

colleagues

other

and

Department.

Mr.

Grant

work

facilities

Engineering

My thanks

L.

Professor

encouragement

this the

available

Mechanical

to

constant

throughout

guidance making

indebted

am greatly

some of

thesis. Mrs.

M. Macpherson

for

thesis. thanks

my special Kathy

for

her

and appreciation

encouragement

and

C0NTENTS

Page SYNOPSIS ACKNOWLEDGEMENTS CONTENTS I

PART

CHAPTER

I

I

GENERAL

1.0

Introduction

1.1

Scope

1.2

1

1.1.1

The dynamics

1.1.2

Optimisation

9

11

OF MOTION

The

Kinematics

2.2

The

Derivation

2.2.1

Constant

2.2.2

Torque

Associating Pin with

11 of

speed

of

Equations

of

input

the Forces

Motion

13

14 16

input Lagrangian'Multipliers and Torques

18 18

Analysis

Methods

8

control

DYNAMICS

2.1

2.4

and

II

EQUATIONS

2.3.1

7

Thesis

of

II

2-. 3

6

Investigation

of

Layout

1

INTRODUCTION

PART

CHAPTER

INTRODUCTION

Solution

24

No.

Page

CHAPTER

III

3.1

DIGITAL COMPUTER SOLUTION OF MOTION EQUATIONS The

Formulation

for 3.2

CHAPTER

CHAPTER

3.2.1

Constant

3.2.2

Torque

Free

-3.4

Computer

29 33 input

speed

33

input

38 41

Motion

43

Programs

1V .

THE LINEARISED ANALYSIS

EQUATIONS

4.1

The Linearised

Kinematics

4.2

The Linearised

Equations

4.3

Simplication of Motion

4.5

Motion

of

Motion

3.3 .

4.4

29

Solution

Digital

Forced

OF

Equations

of

Resonance

AND RESONANCE 44

44 of

46

Motion

Linearised

the

of

Equations 52 54

AnalysisXc, =0

4.4.1

No damping;

4.4.2

damping; Viscous damping '= p.t

54 of

coefficient

56 57

Results 4.5.1

Resonance

in

the

absence

4.5.2

Resonance

in

the

presence

4.5.3

Remarks

of of

Computer

damping

---57 58

V

STABILITY

ANALYSIS

5.1

Stability

Analysis

59

Programs

60 by

a Perturbation

61

Method No damping

damping,

58

4.6

5.1.1

No.

XC

0

61

Page

I

5.2

5.2.1 5.2.2 5.3

5.4

CHAPTER

of

using analysis method

a

Stability orientated

analysis method

a computer

using.

66 66 68

Results 5.3.1

Stability

in

the

absence

5.3.2

Stability

in

the

presence

5.3.3

Remarks Programs

PART

:

III

Strategy

6.2

Problem

damping

of of

damping

68 69 70

Computer

6.1

70

AND CONTROL

OPTIMISATION

72

OPTIMISATION

72 73

Definition.

6.2.1

The desired

output

73

6.2.2 .

The quality

function

74

6.2.3

The

76

constraints

77

Optimisation 6.3.1

Optimisation

6.3.2

Constraints

6.3.3

6.5

Presence

Stability substitution

PARAMETER

6.4

the

"66

VI

6.3

in

Analysis

Stability damping

Flow

77

routine

79 of

chart

the

computer

algorithm

81

Results 6.4.1

Introduction

6.4.2

Function

6.4.3

Path

Further

80

to

the

81

results

85

generation

89

generation Remarks

and

Computer

Programs

91

No.

Page

CHAPTER

VII

CONTROL

7.0

Introduction

7.1

Conceivable

7.2

7.3

7.4

92 Control

7.1.1

Off-line

7.1.2

Partial-on-line

7.1.3

Adaptive

Methods

93 94

methods

95

methods

95

methods

Analysis

96

7.2.1

Problem

7.2.2

Variational

7.2.3

Direct

definition

96 97

method, search

98

methods

Results

98

7.3.1

Examples:

series

1

99

7.3.2

Examples:

series

2

105

Search

for

A Strategy Linkage

PART

CHAPTER

92

IV

in

EXPERIMENTAL DISCUSSION

VIII

8.0

Aim

8.1

Apparatus

8.2

Description 8.2.1

of

Optimum 106

EXPERIMENT

:

the

AND CONCLUSIONS

INVESTIGATIONS OF RESULTS

Experiment

AND 108

--

--108 109

of

Apparatus-,

Experimental

rig

1

109 109

8.2.1.1

'Mechanical

components

109

8.2.1.2

Electrical

components

ill

8.2.2

Experimental

rig

2

112

No.

' Page'

8.3

Calibration 8.3.

8.4

l..

Linkage

input

8.3.2

Spring

stiffness.

8.3.3

Damping

8.3.4

The

114 114

coefficient displacement

slider

114 114

Experimental

8.4.1.1

8.4.2

1

rig

Comparison of and theoretical

8.4.1.2

IX

113

parameters

Results 8.4.1

CHAPTER

113

Variation oscillation

Experimental

of rig.

8.4.2.1

Slider

8.4.2.2

Resonance

114 experimental results

the with

slider crank

Concluding

9.2

Recommendations

PART V:

119 119

curves

Remarks for

F

APPENDICES

I

117 119

oscillation

REFERENCES. -

REFERENCES

speed

2

CONCLUSIONS AND RECOMMENDATIONS FURTHER WORK

9.1

115

FOR 122

122 Further

Work

P.ND APPENDICES

125

No.

PARTI

INTRODUCTION

1

Chapter

GENERAL

1.0.

INTRODUCTION

Introduction Mechanisms great It

numbers

is

not

of

present

day

to

in

(1 -

function

the

dynamic

must

inputs

that

will

The'second investigating

the

interfere

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with

The

mechanisms.

and the dynamic

resonance characteristics.

third

step the

flexibility

studies

and stability,

include

analysis. parts

and do not

the

the

items

and power

of

neighbouring the

of

study

distribution of

motion.

constitutive

components is

the

and

output

motion

the

or

towards

dimensions

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motion

from

resulting

inertias

in

work

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kinematic

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In

acceptable

step

desired

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motion

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and ensuring

effects

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harmony

first

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generate

step

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therefore

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most

motion.

and have

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in

output work

be efficient

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various

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is,

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mechanism

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synthesis,

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the design

a specified

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find

to

and

characteristics.

designing

industry.

engineering

3).

of

surroundings,

in

present

investigating

function

generate

so,

are

therefore,

ref.

The main

These

in

motion,

mechanisms,

its

types

interest

aspects

doing

various

surprising,

substantial

cases,

of

of

masses

various such

dynamic and

components. as balancing,

transmission

-2-

Linkages

A number

mechanisms.

focused

therefore the

various

their

The

into

two the

constitute

categories

(a)

in

the

work

16)

which,

type

of

desired

means

limit

on the

being the

the

of

number

linkage is

considered)

in

the

values

(five,

in

methods

as solving

define

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the

obtaining

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mechanism,

used

motion"

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describing

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a four-bar to

linkage

define

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configuration

four

points

precision finding

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parameters

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initial

results

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points".

in

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adopted

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if

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achieve

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may be considered

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finding

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output of

example,

parameters

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is

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refs.

points

form

and

:

in

positions

precision

type For

studied.

or

aspects

of

"precision of

number

by

determined

points

of

literature

the

desired

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to

-

Examples

of

The most

motion.

specifying

in

called

and dimensions

output

two

ways

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may be found

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of

studies

kinematic,

analysis.

synthesis

linkage

these

of

concern

studies

type

this

and

part

and kinematic

synthesis

on studying

concerning

major

have

attention

part

of

class

researchers

greatest

The kinematic

method

of

phenomena

mechanism. fall

an important

constitute

the

the

methods

include set

such of

N equations,

sufficient the

different

to

-3configurations motion

output

the

in

in

used

least

Other

others.

in

solutions

inversion

the

of through

point fixing

the

pin

joint

and

form

the

of

(23 -

methods, and direct author

Such

methods

except

in

difficulty

in

obtaining

of

to the

start

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criterion.

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and

dimensions

in

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may be found

in

employed

gradient

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to

a minimum

apply

in

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and

in

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on initial

methods is

the

of

values

methods of

opinion

gradient

rely

are

random

difficult

stationary methods

applying in

22),

ones, because

The gradient

them.

gradient

are

simple

a linkage,

the

and

In

methods.

linkages,

existence

work

programming

search

tracing

parameters

(17 -

The methods

linear

gradient

the

an interest

ref.

methods,

31).

non

been

has

as

such

links.

remaining

problems.

synthesis

techniques

and determining

there

control

optimal

its

of

and

on graphical

taking

of-some

locations

based

and choosing

output

values

Recently

refs.

the

matrix

techniques

mechanism,

are

techniques

different

include

are

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are

again,

which, These

used.

techniques

displacement

methods

the

to

iterative

as Newton-Raphson,

such

are

fit,

an

finding

and

respect

solution.

square

prescribed

as constructing

Various the

achieving

the

generation

with

parameters.

included

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output

values

stationary

mechanism

methods

or

function

error

to

corresponding

located.

error

guesses

descend

along

The direct

-4-

search in

methods

linkage

in

the

for

criterion

minimum

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This

the

progress

parameters that

the problem

however,

greatly the

of

yield

obviously

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efficiency

random

error

linkage

methods

type

choice, and

the

criteria,

the

upon

The

parameters

these

of

a reduction

of

of

of

convergence

depends

studied.

values

set

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error.

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best

values

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values

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choose

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vectors

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error

investigate

methods

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the

of

and

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parameters

value

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space

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linkage

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along

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choice

of

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type

of

method

used. Examples

the

of

be found

in. refs.

consists

of

the

and

displacements equations

are

set

analysis

straight

forward

Studies

them

to

to

in'vector

kinematics. to

from

this

are

dynamics than of

taking

the

kinematic forms

scalar of

used

of those

workers

and examples

and.

complex:

techniques

attention

field

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of

use

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A number

The

obtained

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of

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the

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analysis

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40).

loop

links

techniques

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first

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the

individual

velocities

kinematic

up the

setting

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mechanism of

(32 -

describing

motion

in

work

and in

the

solution.

mechanisms concerning have, of

however. their

work

5_

may be found

in

(41,44)

balancing springs

and

and inertias

to

and

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Hill's

type

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linear

non

torques,

done

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work

forces

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effects

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to

deals

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loci

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pin

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some work in

69)

linkages.

and briefly

on the

pin

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forces

(55 -

the

(65 -

bearings

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refs.

the

clearances of

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in

cases

yield

velocities

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effects

may be found the

concerns

ref.

dynamic

most

to

motion;

response,

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accelerations.

in

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of

equations

forces,

pin

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forces,

dynamics

standard

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masses

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linearised

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54), in

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values

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of

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equations

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distribution

minimum

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adding

the

yield

studied1refs.

the

69).

concerns finding

moments

mechanisms

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ref.

shapes in

minimise

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of the

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mechanisms. a five

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two

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motion.

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loop

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the

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64),,

-6-

inputs-

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mechanism

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of

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dynamics.

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plot

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to form

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Mathieu

when

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independent

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written

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of

optimum minimum

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instability

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The value

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Optimisation

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finding

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method

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rocker-to-

introduced. manner

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This to

-9acceptable

becomes within

control

of

this

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Alternatively

output. the

pivot

under

his

the

supervision the

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steady

state

motion

spring

rates

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is

rig

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to

input

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crank

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rates.

spring

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results.

having

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different

theoretical

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from

obtained

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thesis

of

introduction;

Part

first

the

contains

the

second

the

third

analysis. deals with

the

II

Part

of

analysis III

with

parameter

of the

the

layout

includes

the

general

four

of

equations equations

of

the

fourth

of

two

chapters,

optimisation IV

consists

the

chapters,

and

consists

Part

parts,

of

derivation

Part

control.

I

consists

solutions

resonance

five

into

as follows.

is

which

divided

is

The thesis

second

for

investigate

for

investigation finding

concerns

the

built

second The

slider

the

analysis.

Layout

first

the

the

to

also

are

(53).

linkage

the

value

theoretical

of

used

are

the

constant

with

with

coefficient

used,

and

rig,

crank

them

relative

results

author ref.

out

second

1.2.

by

built

carried

also

rigs

investigations

Two experimental is

by proper

pivot.

Experimental

first

output

achieved

may be

area

an output

a desired

and

accessible

defined

1.1.3.

the

releasing

of

as a , result area

in

errors

motion, motion,

stability

and the of

two

the

-

chapters,

the

investigations Part

V contains

first

10 -

contains

and the

the

second

references

the

experimental

the

conclusions.

and appendices.

I

FART''

II

"DYNAMICS

5

1

- 11 CHAPTER

OF MOTION

EQUATIONS

the

Consider is

degree

a two

being

the

crank

slider

the

driving

force

F2 acting

force

is

by u.

inertias

are

2.1.

two

as

The

kinematics

£2CosO2

link

displacement

force or

masses

in

the

figure

and

origin

at

may be defined along

are F1

constraint

dimensions,

link

of

forces

The viscous

slider.

It

degrees

coefficient

equations Li,

the

loading

controller

X and Y with

Kinematics

the

(2.1).

by F3 and the

shown

The

constraint

ground

on the

The

are

coordinates

the

p and

fig.

external

the

T1(02),

represented

damping

and

imposed

The

in

mechanism,

"02"

angle

torque

on point

acting

shown

freedom

of

"u".

the

linkage

II

the

damping of and reference

A.

by writing

the to

and perpendicular

the

i. e.

+ Q3 CosO3 -

R4 Cos(04

+ a)

-u

Cosa

=0

(2.1) R2SinO2

+ Q3SinO3

-

k4 Sin(04

+ a)

-u

Sinct

=0

0 Ü

w

0

z

N 0) U-

12 -

-

In input

in

addition,

w

the

of

the

case

a constant

of

a further

crank,

speed

equation

the

of

form 02 is

required.

in

suffix

The

= 1,2,3.

The

constraint

notation

displacement

in

expressed

equations

(2.3) 03

X2.3)

= 0,

from

¬591E3,041d

Equation angles

represented

(2.3)

terms

of in

and written

as,

vi (02,03'04ºu,

the

be

as

may be derived

velocities

may

equations

(021031041u)

-j

suffix

(2.2)

=0

above

notation

fý

where

wt

and

be

can 04

)=0

solved

to

explicitly

functions

as

Similarly

u.

(2.4)

of

the

angle

angular

02

give

and

velocities

"r

from

03 and 04 may he obtained positions

and

accelerations derivative that

of

equation

kinematic

speeds

of

crank

(2.4) and

may also

be obtained

equation

(2.4).

(2.3)

solution

is of

sufficient the

linkage.

as functions The

slider. by taking

It

is, for

of

the

therefore, the

complete

first clear

13 -

-

2.2.

The Derivation The

as

follows

ýt

DT ýq'

of

equations

Lagrangian

Equations

of

motion

multipliers

the

using

may be stated

This

:-

i

T is

the

total

and qi

are

the

generalised

Qi

generalised

-3T

the

derived

are

method.

= Q. + En 1i -q =1 i.

where

Motion

of

conservative

Constraint

equations.

the

the

of

coordinates

qi

conservative

and non

of-constraint

number

multipliers

forces

system

and velocities,

including

Lagrangian

generalised

(2.5)

energy

n the

effects, ai

The

kinetic

forces

equations,,

8±. . . aql

A.

and

Qi may be written

fi

the

in

the

form Qi

ate)

(g.

= mj.

ri

+ Fi

.

Dqi where

the

vector, application of

masses,

coordinate the

of

(2.61

Ta

3

aqi

g the

vector

the

of

forces

external

gravitational

Fi

point

of

and Ti

the

vector

torques. Let

u.

the

Now in

these must

Dqi

the

represents

m

+

generalised

order

found. of

derivative.

This velocities.

the

the

velocities

These

y coordinates

following

obtain

the

coordinates be

to

coordinates

can

kinetic of

of for

the

masses centres

be 02,03,04 energy

be obtained

centres

yields,

qi

in

centres

terms of

by writing and

and

taking

C3 and G4,

of

masses the

x and

the

first

the

14 -

-

ýSCZ 02+2.

ýýG3 = ýü VG4

Hence

G3032+22G3.

the

total

l

-2RG4 . SinO4.04

kinetic

may be expressed

. uý J

energy

of

the

mechanism

as

2 022 IA . 2

T=

(2.8)

222 +lCG404

=

Ö3p3

k2Cos (02-03-v3)

IG3.0632

IG4.042

+2

+2

2

m5ü

(2.9) 1212 +2m3VG3+2m4 in

Finally,

and written

coordinates

of

xp

the

forces

that

order

be expanded

external

VG4

for

points

must

= £2 cos(02+ai)

(2.5)

equation linkage

this of

the

application

be known.

may

the

of

These

are

+ d1cos(03+a3+a1)

yP = £2 sin (02+a2)

(03+a., +al)

+ cl1sin

(2.10)

and XD = Qlcosal'+ YD -

ß1sina1

viscous

2.2.1

sin(a+a1)

joints,

pin

and no frictional force

coefficient written

+u

ideal

Assuming forces

u cos(a+al)

u the

of for

forces on the

acting

following

Constant

Speed

a

a damping

motion

may be

cases.

Input.

i. e., =

of

from

with

slider

two

a constant

02

apart

equations

the

Assume

gravitational

wt

crank

angular

speed,

w,

15 -

-

is

This equation

in

addition

replaces

the

need

for

the

torque

Lagrangian later,

for

In

itself

of

equations

fact

the

leads as

to

a be

will

shown

torque.

and Fi

are

the

values

x and y directions may be written

motion

and

an expression it

external

Fi

that

F1 in

force

(2.1)

A3 which, the

constraint

equations

specifying

T1.

Assuming the

to

multiplier is

as a further

considered

of

the as follows

:

m3 R2 !CG3Cos (0 2-0 3-v 3 ). 0 3 +m3Q2R,G3 Sin (0 2-0 3Iv 3 )-0 3 +X1 !C2SinO 2a2R2

Cos® 2a

+m 3 2gR

(IG3+m3ZG3) 03-m3 £2 £G3Sin (02-03-v3)

1) (02+a1)=O

02 +A1£3Sin03 (C13+a3+a1)

(03+v3+. a1)+Fid1Sin -x 2Q3CosO3+m3gQG3Cos

(2.12)

)=0

-F1'"d1"Cos(03+a3+a1

(IG4+m42

(1.11)

Cos (0 2+a G2

(02+a1)-F1yz2Cos

+ m3g£2Cos (02+a1)+F1X£2Sin

04 m4 tG4Sin04 4)

+X224Cos (04+a)+m0ZG4Cos

2

(04+cc) Sin -J119,4 . (04+a+a1)

(2.13)

=0 2

(m +m ) ü'-m 454

Z

+X Sina+mp-,

d' k CosO SinO . 44.0 G4 G4 444 -m Sin (a+('l)-F2+ul

ý:

vý

=O

+71 Cosa .1 (2.14)

16 -

-

(2.11)

Equations second

order

Together are

differential

non-linear

by

obtained

following

the

with

(2.14)

to

through

taking

equations.

two

the

are

which

equations derivatives

second.

of

N

(2.1)

equations a2,

al,

they

+A,3Cos03.632

"022

23CosO3.03

input the

Sin®

-L

case

Assume

an input

as in

of

of

forms

the

(0

T1 = T1 b

= 0"

(2.16)

torque

the the

an induction

(2.15)

ü"Sina

may be a function

a function

in

expressed

u

+Q4Sin (®4+a) "O42 =03

3

Input

position, or

62

Torque

This

angular

3,

ä4,

-Q4Cos(04+a)"042

-

-Q4Cos(04+a)0o

Q2SinO2 ". 022

crank.

0

' + ü' Cosa 0! -

Sin(04+a)

-Q4

+Q2Cos02

2.2.2

for

solved

.I

X3.

and'

P, 'SinO3.03

-

be

may

""

case crank

motor.

Tl

acting

of

the

of

on the crank

a torsion

angular These

bar

speed,

as in

may be

of (a)

2)

(2.17) and

(02)

T1 = T1

(b)

m respectively.

It

is

clear

that,

in

both

cases,

the

17 -

-

constraints in

are

(2.1).

specified

of

equations

(2.5)

force

forces

on the

acting

are

equations

-x291 2Cos02

(2.13) the

set

simultaneously

these

two

of with

four

02-m32.2 9, Sin (02-03-v3)02L G3

equations

are

exactly.

equations

equations of

= 0. (2.18)

equations

(2.19)

=0

to

similar

As mentioned

may be solved obtained (2.1).

from In

the this

case

:

R2SinO2.02+Q3Sin030ý3

+Q2Cos02 .

Two of

d1. Cos(03+a3+ai)

(2.14)

and

derivatives are

four

C3Cos(03+v3+a1)

Sin(03+a3+ai)-Fly.

The remaining

second

of

Sin(02+a1)-Fik2Cos(02+a1)

Q3Sin03-A2QJCos03+m3g,

+ F1X. di.

before

a set

the

- T1 + m2gZC2Cos(02+a1)

03+m391 91 ) Z Cos (02-03-v3) +M G3 3 G3 3 G3

equations

from

:

+ m3gR2Cos(02+a1)+F1XR2.

+ ýl.

apart

+M391291G3Cos(02-03-v3)03+M3Q2ZG3Sin(02-03-v3)032

+ ý1912Sin02

(1

slider,

and

ideal

may be obtained.

equations

(IA+m3Z22)02

forces,

no frictional

and

differential these

equations

gravitational

assuming

joints

viscous

two

By application

as before pin

the

only

d4 +ü' Cosa (04+a) - . 4Cos

02+Q3 Cos 0303

214Cos (04+a) 04' = 01

(2.20)

18 -

-

A. 2Cos02.02

+2 3Cos03.03

-Z2Sin02.0c22

2.3.

Pin

Forces

In

and

the

time

and

Lagrangian

formulation

The Lagrangian formulation

and

larger

forces

the

between

forces

are are

accelerations of

equations

available

dealing The

large

of

as will

two

is

the

be seen

later

Once

possible.

the

of

Lagrangian

include

not

this

remaining

obtainable from

number

The

but

all

hard

computer

and

does

easily

of

expense

matter.

accomplished

the

equations.

solution

the

compact.

facilitates

differential methods

the

and more

the

to

usually

However,

at

explicitly

is

association internal

but

of

make

an association

2.3.1

more

with

approach

is

motion

method

unfortunately,

multiplier,

Newtonian

easier

a straightforward

equations

internal

is

numerical

wear

Multipliers

consuming.

numbers

day

of

multiplier even

soft

the

equations

cumbersome

(2.21)

= 0.

Torques.

many mechanisms

formulating

present

Lagrangian

the

u Sine

+R4Sin(04+a)042

-Q3Sin030r32

Associating

with

04 (04+a) -RQCos

since

the

the

solution

of

the

motion.

Analysis

The number be expressed h=3

in

the

of

lower

pairs

in

a linkage

may

form (N-1)-n 2

(2.22)

19 -

-

where

N is

degree

of

the

freedom

constitute number

lower

of

constraint of

the

fj(gl,

and where

are

q2"""yqk) the

are

qk

j=

m be number

2m and

the

If

constant

of

equations

the

number

of

are

obtained.

Each a pin

constitute

the to

remains

lower the

pairs joints

pair

force input

and

contain

input the

A's

the

links

input

Fig (2.2)

will

inputs the

are

k multipliers

with

torques.

multipliers

2m multipliers

Therefore,

to

the

Z is

where

(2m + £)

the the

in

of

number

(2m + Q),

torques.

associate

and

and

equations

motion

hence of

a linkage

then

then

and

only).

coordinates

becomes

inputs,

in

linkage

the speeds

constraint

constitute

of

equations

coordinate

constraint

generalized

2m murtipliers. forms

the

two

linkages

loops

of

a

yields

each

(planar

system

. linkages

containing

loop

along

one

the

All

each

Each

pairs.

equations,

Now let

loops

more

reference

and n is

linkage.

the

of

or

one

links

of

number

get

the

the

it

only

corresponding

forces

at

-

The values of the

linkage

the

constitutive

when

function

the

As an example

of

02 and 03 or

04 and 03.

four

lead

Both

to

motion.

(2.2)

are

may be

forms-of

Therefore

it

forms

motion

by

considering

the

in

of

the

forms

shown

of

linkage in

is

possible

equations of

fig.

fig.

(2.3).

(D

G)

(2.3) system

v Alternative for

purposes

treatments of

as a

essentially

different

different

of the

as a function

to

-Pig.

coordinates

fig.

equations

0

around

in

of

any

choice

looping

of

specifying

of

but

obtain

the

upon

link-3

of

equivalent

depend

links.

energy

calculated

ai

and direction

coordinates

kinetic

of

20 -

analysis.

of

of (2.2)

21 -

-

This

Each

obtainable. force

at

as to

to the

at

proof

(2.2)

in

two

configurations

as

coupler,

joint

constraint formulate

two

Al

-rocker

forces the

pin

the

pin broken. (2.3.

b)

A2 the

of

is

in

argument

above

linkage

the

exactly

shown

in

and

components

Consider but

is

are

pin

joint.

of

as follows.

stated fig.

the

the

fig.

of

multipliers

coupler-rocker

A simple

to

system

Xi

of

]inkage,.

the

where

of

values

corresponds

set

example,

values

corresponding force

an

of

sets

joint.

the

Therefore leads

four

that

means

force.

Fig(2.4)

of

equivalent fig.

(2.4) by

replaced

fig.

may he

(2.4.

b).

where

the

four These

forces

22

-

Now let

fig.

(2.4.

for

a. )

fig.

a torque fig.

02

= wt.

DT

a

a)

as

Euler-Lagrange

assume is

an

-

method

a constant

crank

treated

in

(2.4.

this

constant

fig.

b)

speed

constraint

extra

aT

= Q. +

X.

as

and

equation

DT

+F

1

]c

ar

+ T-30

fi

and

a

are

Therefore

if

aý

=

aqi

the

aqi

in

in

positive

02 = wt.

similar

for

forces the

(2.24)

method

(2.4.

b) along x,

y

loop-constraint

two

equation

(2.24)

and

terms.

are

the

plus

equations

Euler

Dqi 2agi

where Fk are the four constraint r. coordinates -their . respective directions

(2.23)

aqi = Q.

qi

A- method

af.

aq i

i

(2.23)

for

We obtain

-'

aq.

at

and

maintaining

(2.4.

a aT at aq"i

b)

motion

method

basic

the

co which

T2

in

multiplier

and

(2.4. input

speed

the'equations"of

us write

Lagrangian

the

using

-

apart

Equations from

the

last

j' = 1,2 (2.25)

af

and

"

30

t

aqi

2

for

j

for

j=

=3

aqi

Then, 7ý =F j j

1,2 (2.26)

A.

and i. e., forces

the

= T2

multipliers

components

for

j=

3

are

equal

to

and the

input

torque.

the

internal

23 -

-

The

k2SinO

Y_ c31 and

forces

of xc

(2.4.

because;

by are

8ql

three

Hence

added.

-Q2Sin02

and

Q2CosO2 and

3 =1 a02

a0

a021

302

9,3 Sin03

and

ý0

and

8x1 c 303

af_

=0 803

C two

yields Thus

three by

R2 SinO2.

=-

Q2 Cos02

=

Last

terms

1st

equation

in of

motion.

=1

Dxcl

8fl

8f

c1

pair

freedom

a0

of

a03

(2.23)

of

ay

of

DO 33

lower

(2.24). similarly a and we have 8qj. aX

302 -

degrees

the

releasing

and a

has

b)

of a012

:

Sin0

Now fig.

expanding

C2 are

(Q1+Q4CosO)4

Y-L c44 1

equations

:

SinO

2+Q3

on

=-2

degrees

on C1 are

acting

+ 2.3CosO3

22CosO2

o=1

forces

of

coordinates

P3 Ccs03

= -t3

=

8f a03

=O

I- ) Sin03

k3 Cos03

Last

terms

2nd equation motion.

in of

24 -

-

of 1=

2'4 SinO4

DO 44)

axC2

and

SinO = 1C 4

a0 ayc2

aft DO 44

Cos04

=-R4

and

Cos0 4

=-k4

a0

in

Last

terms

3rd

equation motion.

of

af3

af3 O

_

a04

i. e.,

= G.

a04

x1 = F43

A2 = P43

'

Since

is

(2.25)

equation

the

and therefore

true

A3 = T2.

and

velocities

accelerations,

positions

are

given

from

remaining

pin

forces

at

the A,

and the

solution,

D may be easily

B and

obtained. METHODS OF SOLUTION

2.4

The highly

stated

order

differential

in

equations that

the

the

be the

will

four-bar

linkage

be obtained

for

obtained

further

obtained

solution this

pivot;

the

of

can

only the

Therefore,

even

if

linearised

a solution

can

only

been

be seen

simplifying

in

a computer.

case.

a particular

as will

the

bearing

then

the

no movable

day

linearise

possible

of

are

analytic

present to

the

of

results

with

using have

equations

u is

variable

coefficients

equation

with

An attempt

methods.

earlier

generalised task

an impossible

is

mathematical

if

motion

A straightforward

solution

cases,

second

non-linear

equations.

mind

of

equations

later

in

assumptions

Ilowevor, chapters are

in IV

some and V,

made a

be

25 -

-

lincarised which

with

equation

can

a near

be obtained

using

computer.

The

efficient;

however,

either

the

digital

computer

is

time

sharing

researcher and

the

the

only

The

the

accuracy

and

widely

used

Runge-kutta

to

the

the up and is

by

one

above

digital

sharing

it

was decided

the

a form Z"

dD(t,

Z represents

the

and u and methods

its

depends

choice equations.

can be

earlier (see

to

own

3.1)

Z)

(2.27)

unknowns

Z the are

the

derived

equivalent

_

the

and

motion

of

has

Each

one

the

are and the

methods

of

adopted.

integration

of

series

complexity

and

a numerical

integration

of

methods.

The equations in

of

method

and disadvantages

nature

Runge-kutta

set

be used

IBM 360/67,

methods

methods,

predicted-corrector

02,03,04

hand,

this

Owing

efficiency

The most

where

be

can

other

can

and

solution.

on the

expressed

digital

quick

easily

which

a time

of

depend

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