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Dynamic aalysis of a small weaving loom

Gülen, Serdar

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Dynamic Analysis o f a

Small Weaving Loom

by

Serdar GUlen B.Sc.

Thesis submitted f o r the degree of Master of Science i n the U n i v e r s i t y of Durham.

August, 1976

24 K0Vv-

ABSTRACT

The p r o j e c t deals w i t h an i n v e s t i g a t i o n t o determine e x p e r i m e n t a l l y the

a x i a l and bending s t r a i n s (normal t o the plane o f the mechanism) i n the

two connecting rods of the weaving mechanism of a small t e x t i l e loom.

A complete t h e o r e t i c a l , kinematic, force and s t r e s s a n a l y s i s has been

made on the s i x - b a r chain c o n s t i t u t i n g the mechanism. The peak t o peak

s t r a i n values have been measured a t various d i f f e r e n t crank speeds. The

nature o f the bending s t r a i n s i n a d i r e c t i o n normal t o the plane o f the

mechanism have been f u r t h e r examined by s t a t i c t e s t s which have been

performed on the mechanism. Measured dynamic s t r a i n data f o r the

connecting rods i s presented and comparison i s made between c a l c u l a t e d

and measured values. Experimental r e s u l t s f o r a x i a l peak t o peak and

c y c l i c s t r a i n v a r i a t i o n showed good agreement w i t h the c a l c u l a t e d values.

i i

ACKNOWLEDGMENTS

I wish t o express my sincere g r a t i t u d e t o Professor G.R. Higginson

and Dr. P.A.T. G i l l f o r t h e i r guidance, advice and continuous

encouragement du r i n g the preparation o f t h i s work.

I would also l i k e t o express my thanks t o Mr. T. Sweeting f o r

h i s constant help d u r i n g the experimental stage of the p r o j e c t and t o

Miss J. Campbell f o r t y p i n g t h i s t h e s i s .

I am indebted t o my parents f o r p r o v i d i n g me w i t h f i n a n c i a l

support.

i i i

CONTENTS

Page

Abstrac t i

Acknowledgements . . . . . . i i

Contents i i i

Nomenclature

I n t r o d u c t i o n

1 Stages encountered d u r i n g the

development of the p r o j e c t 1

2 H i s t o r i c a l background t o the

i n v e s t i g a t i o n 2

3 Plane mechanisms i n general,

c u r r e n t research and short

b i b l i o g r a p h i c a l review .. . . . . 3

Chapter 1

De s c r i p t i o n of the machine .. .. . . -6

1.1 Power transmission system .. . . . . 6

1.2 Mechanism Unit box 8

Chapter 2

Kinematic a n a l y s i s of the mechanism

2.1 I n t r o d u c t i o n t o kinematic a n a l y s i s

of the mechanism .. .. . . 22

2.2 Geometry o f the mechanism .. .. .. 22

2.3 V e l o c i t y and a c c e l e r a t i o n

Analysis - A n a l y t i c method 24

2.4 Algebraic approach 36

2.5 A p p l i c a t i o n of Raven's An a l y s i s 48

i v

2.6 Graphical Approach 54

2.7 Energy v a r i a t i o n of the system 55

Chapter 3

Force and Stress Analysis

3.1 K i n e t o s t a t i c Approach 58

3.2 A p p l i c a t i o n of v i r t u a l work

method t o the mechanism 65

3.3 The Power equation .. .. .. 66

3.4 Graphical approach 70

3.5 S i m p l i f i e d Force A n a l y s i s 73

3.6 Stress Analysis 75

Chapter 4

Instr u m e n t a t i o n , Experiments and

experimental Results

4.1 Instrumentation 86

4.2 Elements of the experimental

set up 90

4.3 S t r a i n gauges and gauge

i n s t a l l a t i o n s 91

4.4 Experiments

1 Dynamic s t r a i n measurements 94

2 S t a t i c bending-strain

measurements .. .. .. 95

3 Stress d i s t r i b u t i o n i n

l i n k s 3 and 5 95

4 T e n s i l e Test 99

4.5 Dynamic s t r a i n measurement

r e s u l t s 103

V

4.6 Speed F l u c t u a t i o n 103

Table 4.1 105

Table 4.2 116

Table 4.3 128

Chapter 5

Discussion o f Results

5.1 Determination of A x i a l

working stress f o r l i n k 5 -

A t h e o r e t i c a l approach 145

5.2 T h e o r e t i c a l and experimental

s t r e s s range v a r i a t i o n w i t h

the crank speed i n l i n k 5 151

5.3 C a l c u l a t i o n of e c c e n t r i c i t y

from experimental data, f o r

l i n k s 5 and 3 154

5.4 E c c e n t r i c i t y - dynamic case 156

5.5 Determination of A x i a l working

stress f o r l i n k 3 - a theor­

e t i c a l approach \ 157

5.6 T h e o r e t i c a l and experimental

a x i a l s t r e s s range v a r i a t i o n

w i t h the crank speed i n l i n k 3 158

5.7 Bending stresses i n a d i r e c t i o n

normal t o the plane o f the

mechanism 159

5.8 Stress d i s t r i b u t i o n I n the l i n k s 160

Table 5.4 t o 5.7 1 6 1 - 1 6 4

5.9 General Considerations 168

v i

Chapter 6

Conclusion 169

Appendix ( A l ) 171

Appendix (A2)

Appendix (A3) "'186

References , . . . . . . 187

v i i

L i s t o f Figures, Tables and Plates

Page

Fi g . 1.1 7

Table 1.1 s p e c i f i c a t i o n of l i n k s 8

F i g . 1.2 Mechanism 9

Pla t e 1.1 Power transmission system 10

Pl a t e 1.2 Mechanism u n i t box l e f t end 10

F i g . 1.3 Mechanism shown as p i n j o i n t e d rods 11

Pl a t e 1.3 Mechanism box r i g h t end 12

Pla t e 1.4 Mechanism complete w i t h connecting rods 12

Pl a t e 1.5 Connecting rods, r i g i d t r i a n g u l a r l i n k 4

and l i n k 6 13

Pla t e 1.6 Main d r i v e s h a f t 13

Table 1.2 s p e c i f i c a t i o n o f s h a f t s 14

F i g . 1.4 Main d r i v e s h a f t crank arm end d e t a i l e d

drawing 15

F i g . 1.5 Link 6 and s h a f t 0 w i t h the comb 16

F i g . 1.6 r i g i d t r i a n g u l a r l i n k 4 17

F i g . 1.7 S i m p l i f i e d weaving mechanism. H o r i z o n t a l

p o s i t i o n i n g o f sh a f t s 0 , 0 o and 0 18 1 A o

P l a t e 1.7 Top view - mechanims u n i t box 20

Pl a t e 1.8 s h a f t 0 , s h a f t bearings and t h e bevel

gears 20

Pla t e 1.9 Needle arms 21

Pl a t e 1.10 s h a f t 0 , comb and bearings 21

F i g . 2.1 23

F i g . 2.2 V a r i a t i o n of angle gama w i t h the crank angle 25

v i i i

F i g . 2.3 V a r i a t i o n of angle beta w i t h the crank

angle 26

F i g . 2.4 V a r i a t i o n of angle upa w i t h the crank

angle 27

F i g . 2.5 V a r i a t i o n of angle xlamb w i t h the crank

angle 28

F i g . 2.6 V a r i a t i o n of transmission angles c a l and

yum w i t h the crank angle 29

F i g . 2.7 A n a l y t i c a l v e l o c i t y and a c c e l e r a t i o n

a n a l y s i s 35

F i g . 2.8 V a r i a t i o n of angular a c c e l e r a t i o n of

l i n k 3 w i t h the crank angle 37

F i g . 2.9 V a r i a t i o n of angular a c c e l e r a t i o n o f l i n k

4 w i t h the crank angle 38

F i g . 2.10 V a r i a t i o n of angular a c c e l e r a t i o n o f l i n k

5 w i t h the crank angle 39

F i g . 2.11 V a r i a t i o n of angular a c c e l e r a t i o n o f l i n k

6 w i t h t h e crank angle 40

F i g . 2.12 Angular v e l o c i t y curves 41

F i g . 2.13 V a r i a t i o n o f angular v e l o c i t y r a t i o w i t h

the crank angle 42

F i g . 2.14 (a-b) 44

F i g . 2.15 Dead-centre p o s i t i o n s f o r l i n k s 4 and 6 45

F i g . 2.16 49

F i g . 2.17 V a r i a t i o n o f k i n e t i c energies o f s h a f t s

0„, 0' and l i n k s 3 and 4 w i t h the crank

angle 56

F i g . 2.18 V a r i a t i o n of the k i n e t i c energy w i t h the

crank angle 57

X

P l a t e 4.1 88

Plate 4.2 88

P l a t e 4.3 E l e c t r i c motor and speed v a r i a t o r u n i t 89

F i g . 4.3 S t a t i c b e n d i n g - s t r a i n measurement 96

F i g . 4.4 S t a t i c b e n d i n g - s t r a i n t e s t f o r l i n k 5 97

F i g . 4.5 S t a t i c b e n d i n g - s t r a i n t e s t f o r l i n k 3 98

F i g . 4.6 100

F i g . 4.7 101

F i g . 4.8 Tensile t e s t f o r l i n k s 3 and 5 102

Table 4.1 Speed f l u c t u a t i o n 105

F i g . 4.9 T h e o r e t i c a l and experimental v a r i a t i o n

of a x i a l s t r e s s range i n l i n k 5 w i t h the

crank speed 106

F i g . 4.10 Experimental v a r i a t i o n o f bending s t r e s s

range i n l i n k 3 w i t h the crank speed 107

F i g . 4.11 Experimental v a r i a t i o n of bending s t r e s s

range i n l i n k 5 w i t h the crank speed 108

F i g . 4.12 T h e o r e t i c a l and experimental v a r i a t i o n

of a x i a l s t r e s s range i n l i n k 3 w i t h the

.crank speed 109

F i g . 4.13 V a r i a t i o n o f peak t o peak stresses i n

l i n k s 3 and 5 w i t h the square o f the

crank speed 110

F i g . 4.14 Experimental s t r a i n p a t t e r n s I l l

F i g . 4.15 Experimental and t h e o r e t i c a l v a r i a t i o n

of a x i a l s t r e s s i n l i n k 3 w i t h the crank

angle 114

F i g . 4.16 T h e o r e t i c a l and e x p e r i m e n t a l . v a r i a t i o n o f

a x i a l s t r e s s i n l i n k 5 w i t h the crank angle 115

i x

F i g . 3.1 62

F i g . 3.2 Free body diagrams 63

F i g . 3.3 V a r i a t i o n of torque t r a n s m i t t e d by l i n k 4

w i t h the crank angle 67

F i g . 3.4 V a r i a t i o n of torque t r a n s m i t t e d by l i n k 6

w i t h the crank angle 68

F i g . 3.5 V a r i a t i o n of Power t r a n s m i t t e d by l i n k s 4

and 6 w i t h the crank angle 69

F i g . 3.6 Graphical f o r c e a n a l y s i s by instantaneous

centres method 71

F i g . 3.7a 74

F i g . 3.7b V a r i a t i o n o f a x i a l stresses i n l i n k s 3 and

5 w i t h the crank angle 76

F i g . 3.8 T h e o r e t i c a l v a r i a t i o n of a x i a l s t r e s s

range w i t h the square o f the crank speed 77

F i g . 3.9 Connecting rod - l i n k 3 79

F i g . 3.10 Link 5 80

F i g . 3.11 V a r i a t i o n of the e x t e r n a l torque T g

required t o d r i v e the mechanism w i t h the

crank angle 81

F i g . 3.12 V a r i a t i o n o f a x i a l f o r c e - l i n k 5 w i t h

the crank angle 82

F i g . 3.13 V a r i a t i o n o f a x i a l f o r c e - l i n k 3 w i t h

the crank angle 83

F i g . 3.14 V a r i a t i o n of t h e o r e t i c a l maximum a x i a l

t e n s i l e and compressive stresses i n

l i n k s 3 and 5 w i t h the crank speed 84

F i g . 4.1 Experimental set-up Block diagram 86

F i g . 4.2 87

x i

Table 4.2 Dynamic s t r a i n measurement r e s u l t s 116

Table 4.3 S t a t i c b e n d i n g - s t r a i n measurement r e s u l t s 128

F i g . 5.1 146

F i g . 5.2 147

F i g . 5.3 , 150

Table 5.1 152

Table 5.2a and

5.2b 156

Table 5.3 . 159

Table 5.4 161

Table 5.5 162

Table 5.6 163

Table 5.7 , 164

F i g . 5.4 165

F i g . 5.5 '.. 166

F i g . 5.6 167

Fi g . 5.7 167

Appendix A3 186

x i i

Nomenclature

• = p s i

0 = Beta

Y = gama

x\ = xlamb

upa

<- AAT

AAR

*B = ABT

AB = ABR

A B f t = ABAT

*L = ABAR »l = ADR V = ADT *^ = ADCR DC A* = ADCT DC

= ADCT

On -2 0/ = 3 ANGC3

"4 = ANGC4

a -5 ANGC5

a = ANGC6 6

crank angle i n degrees - i n a d i r e c t i o n counterclockwise from

the h o r i z o n t a l

angle i n degrees between l i n k 3 and the h o r i z o n t a l , i n a

counterclockwise d i r e c t i o n from the h o r i z o n t a l

angle i n degrees between the side O C of the r i g i d t r i a n g u l a r

l i n k 4 and the h o r i z o n t a l i n a d i r e c t i o n counterclockwise

from the h o r i z o n t a l

angle i n degrees between l i n k 5 and the h o r i z o n t a l i n a

d i r e c t i o n counterclockwise from the h o r i z o n t a l

angle i n degrees between l i n k 6 and the h o r i z o n t a l i n a

d i r e c t i o n clockwise from the h o r i z o n t a l / 2

Tangential a c c e l e r a t i o n component o f A-m/s 2

Radial a c c e l e r a t i o n component of A-m/s 2

Tangential a c c e l e r a t i o n component o f B-m/s 2

Radial a c c e l e r a t i o n component of B-m/s , 2

Tangential a c c e l e r a t i o n component o f B w i t h respect t o A-m/s 2

Radial a c c e l e r a t i o n component of B w i t h respect t o A-m/s , 2

Radial a c c e l e r a t i o n component of D-m/s 2

Tangential a c c e l e r a t i o n component o f D-m/s 2

Radial a c c e l e r a t i o n component of D w i t h respect t o C-m/s 2

Tangential a c c e l e r a t i o n component of D w i t h respect t o C-m/s / 2

Angular a c c e l e r a t i o n of l i n k 2 - rad/s 2

Angular a c c e l e r a t i o n o f l i n k 3 - rad/s , 2

Angular a c c e l e r a t i o n o f l i n k 4 - rad/s 2

Angular a c c e l e r a t i o n of l i n k 5 - rad/s 2 Angular a c c e l e r a t i o n o f l i n k 6 - rad/s

x i i i

VA ' v e l o c i t y o f A-m/s

VB = VB v e l o c i t y o f B-m/s

V : = VBA v e l o c i t y of B w i t h respect t o A-m/s

VD = VD v e l o c i t y of D-m/s

VDC : = VDC v e l o c i t y o f D w i t h respect t o C-m/s

^ = OMEGA2 angular v e l o c i t y o f l i n k 2-rad/s

(JU = 3 OMEGA3 angular v e l o c i t y of l i n k 3-rad/s

00 = 4 OMEGA4 angular v e l o c i t y o f r i g i d t r i a n g u l a r l i n k 4-

U) = 5 OMEGA5 angular v e l o c i t y o f l i n k 5-rad/s

U5 = 6 OMEGA6 angular v e l o c i t y of l i n k 6-rad/s

I n

Mass moment of i n e r t i a o f the corresponding

t h e i r mass centres - kg - m

where- n = 2,3,4,5,6 2 I Mass moment of i n e r t i a of the corresponding s h a f t s - kg - m on

where n = 1,2,3 2 2

EN1 K i n e t i c energy due t o r o t a t i o n - l i n k 4 - kgm /s 2 2

EN2 K i n e t i c energy due t o r o t a t i o n - shaf t 0 - kgm /s 2 2

EN3 K i n e t i c energy due t o r o t a t i o n - sh a f t 0 - kgm /s 2 2

EN4 K i n e t i c energy due t o r o t a t i o n - l i n k 3 - kgm /s 2 2

EN5 K i n e t i c energy due t o r o t a t i o n - l i n k 5 - kgm /s . 2 2

EN6 K i n e t i c energy due t o r o t a t i o n - l i n k 6 - kgm /s

RT1 (Angular v e l o c i t y o f l i n k 3)/(angular v e l o c i t y of crank arm)

RT2 (Angular v e l o c i t y o f l i n k 4 )/(angular v e l o c i t y of crank arm)

RT3 (Angular v e l o c i t y o f l i n k 5)/(angular v e l o c i t y of crank arm)

RT4 (Angular v e l o c i t y o f l i n k 6)/(angular v e l o c i t y of crank arm)

CAL Angle of transmission i n degrees - angle ABO

YUM Angle of transmission i n degrees - angle CDO

1

I n t r o d u c t i o n

1 Stages encountered d u r i n g the development of the P r o j e c t

The primary o b j e c t i v e was t o determine both t h e o r e t i c a l l y and exper­

i m e n t a l l y the a x i a l forces and t h e corresponding stresses i n two o f the

link a g e members o f the mechanism. For t h i s purpose p a i r s o f s t r a i n gauges

were placed on opposing faces o f the rods i n a c o n f i g u r a t i o n capable of

measuring a x i a l s t r a i n s as w e l l as bending s t r a i n s i n a d i r e c t i o n normal

t o the plane o f the l i n k s . Bending s t r a i n s i n the plane o f the linkage

could not be measured because o f space l i m i t a t i o n s and the complex shape

of the connecting rods which p r o h i b i t e d a l t e r n a t i v e p l a c i n g of the gauges.

Two d i f f e r e n t J bridge 120 0 strain-gauge c i r c u i t s were a v a i l a b l e t o measure

a x i a l and bending s t r a i n s independently. The bending strains.were expected

t o be zero. However i n the course of the measurements i t became apparent

t h a t s u b s t a n t i a l bending stesses were present i n these members. To confirm

the dynamic s t r a i n measurement r e s u l t s and t o determine the nature and cause

of the bending s t r a i n s a set o f s t a t i c t e s t s was c a r r i e d out on the mechanism.

The steps taken t o achieve these o b j e c t i v e s were:

1 - A d e t a i l e d examination and d e s c r i p t i o n of the machine

2 - A complete t h e o r e t i c a l kinematic, f o r c e and s t r e s s a n a l y s i s

3 - Preparation of the experimental set-up

4 - Determination of basic m a t e r i a l p r o p e r t i e s f o r the two connecting rods

5 - Measurement of dynamic s t r a i n s

6 - S t a t i c t e s t s on the mechanism aiming t o f i n d out the nature and

cause o f bending s t r a i n s i n a d i r e c t i o n normal t o the plane of

the mechanism

7 ~ Comparison and discussion o f experimental and c a l c u l a t e d r e s u l t s .

2

2 H i s t o r i c a l background t o the I n v e s t i g a t i o n

Bonas Machine Company L t d . , o f Sunderland, England,have s t a r t e d t o

manufacture a high-speed weaving loom t h a t weaves ribbon or tape. The

loom was hoped t o operate a t 4000 r.p.ra. but i t was found t h a t some o f

the moving p a r t s f r a c t u r e d below t h i s speed. By the increase of speed

stoppages due t o component f r a c t u r e occurred more f r e q u e n t l y . At the time

o f the i n i t i a t i o n of t h i s p r o j e c t the loom ran commercially at j u s t over

2000:r.p;m. The basic problem was the short term f a t i g u e f a i l u r e s o f weft

needle arms and the reeds. There were a l s o f a i l u r e s of t h e connectjsng

rods and l i n k s . Although dynamic stresses i n the needle arms and reeds

have been measured experim e n t a l l y and c a l c u l a t e d a n a l y t i c a l l y i n a previous

study ( 24 ) , no i n v e s t i g a t i o n had been made on the l i n k mechanism d r i v i n g

the components. To improve the op e r a t i n g c h a r a c t e r i s t i c s o f the machine, the

designer suggested t h a t the f o l l o w i n g major p o i n t s should be considered i n

f u r t h e r p r o j e c t s ;

a Measuring the s t r a i n o f the components under i n v e s t i g a t i o n under

various c o n d i t i o n s

b I n t e r p r e t i n g the r e s u l t s

c Suggesting p o s s i b l e design changes which would increase t h e

maximum speed of the loom

I n order t o be able t o suggest s o l i d changes i n design, the d r i v i n g

mechanism, weaving components, l u b r i c a t i o n system and power transmission

mechanism should be st u d i e d i n great d e t a i l a n a l y t i c a l l y and experimentally

i n v o l v i n g f a t i g u e , m a t e r i a l , s t r u c t u r a l , dynamic and economical analyses

w i t h d i f f e r e n t models.

3

3 Plane Mechanisms i n G e n e r a l , c u r r e n t r e s e a r c h and s h o r t b i b l i o g r a p h i c a l

review

I f a l l p o i n t s of the curves of motion of a l l the l i n k s of a mechanism

l i e i n one and on the same plane, the mechanism i s c a l l e d a plane mechanism.

A mechanicsm can simply be d e f i n e d as a combination of machine elements

arranged to a c h i e v e a c e r t a i n motion. S i n c e i t d e a l s w i t h the composition

of members of a machine i n t o an assemblage to perform a t a s k , to produce

a new and unusual r e s u l t , mechanisms a r e one of the most f a s c i n a t i n g t o p i c s

i n the f i e l d of mechanical e n g i n e e r i n g . Machine design i s a c r e a t i v e a r t

i n v o l v i n g the p o s s e s s i o n of c a r e f u l a n a l y t i c a l a b i l i t y , good judgement and

a broad e x p e r i e n c e . The b a s i c f a c t o r s which must be taken i n t o c o n s i d e r a t i o n

i n g e n e r a l machine design a r e : u t i l i t y , s a f e t y , c o s t , s t r e n g t h , r i g i d i t y ,

d e f l e c t i o n , f r i c t i o n , l u b r i c a t i o n , wear, heat, n o i s e , f l e x i b i l i t y , c o n t r o l

and appearance. A tremendous amount of work has been p u b l i s h e d on v a r i o u s

a s p e c t s of mechanisms e s p e c i a l l y i n the l a s t two decades. The i n t r o d u c t i o n

of computers i n t o design work has a c c e l e r a t e d the r e s e a r c h a g r e a t d e a l .

Numerous new methods have been developed. Today the techniques f o r s t u d y i n g

the dynamics of mechanisms can be c l a s s i f i e d a s k i n e t o - s t a t i c or time

response approaches ( 4 ) . Current r e s e a r c h t o p i c s i n the f i e l d a r e

concentrated on:

( a ) Optimum mechanism d e s i g n combining k i n e m a t i c and dynamic f o r c e

c o n s i d e r a t i o n s .

( b ) S y n t h e s i s of l i n k a g e f u n c t i o n generators by means of mathematical

methods, models and computers.

( c ) Shaking and b e a r i n g f o r c e o p t i m i z a t i o n .

(d) Experimental and t h e o r e t i c a l study of connection f o r c e s and

frequency response c h a r a c t e r i s t i c s .

4

Dynamic e f f e c t s i n mechanisms become important as o p e r a t i n g speeds

i n c r e a s e and as l i g h t low power consuming economical designs a r e sought.

I n the design and experimental examination of the s t r e n g t h of mechanism

l i n k s the s t a t e of s t r e s s and s t r a i n has t o be i n v e s t i g a t e d . S t r e n g t h

may be checked both t h e o r e t i c a l l y and e x p e r i m e n t a l l y , however i t i s

u s u a l l y i m p o s s i b l e to c a l c u l a t e s t r e s s e s t h e o r e t i c a l l y . T h e o r e t i c a l

c a l c u l a t i o n s a r e sometimes too i n a c c u r a t e because a number of premises

and assumptions have t o be made. Most components and members a r e s t r e s s e d

t h r e e - d i m e n s i o n a l l y but w i t h the e x i s t i n g methods of measurement only

s t r e s s e s a t t h e s u r f a c e can be determined and t h e s e do not give an o v e r a l l

p i c t u r e of the s t r e s s d i s t r i b u t i o n . The most expedient way of s t u d y i n g

s t r e n g t h problems f o r mechanism l i n k s i s to supplement t h e o r e t i c a l c a l c u l a t i o n

by experimental data and c o e f f i c i e n t s .

The e f f i c i e n c y of l i n k a g e s i s g r e a t e r than t h a t of any gear or cam due

t o t h e i r s m a l l f r i c t i o n a l l o s s e s and high power.transmitting a b i l i t y . The

four-bar l i n k a g e due to i t s s i m p l i c i t y has been used f o r t r a n s m i s s i o n of

motion i n g e n e r a l . Although i t i s the s i m p l e s t p o s s i b l e l o w e r - p a i r e d

mechanism, s i n c e more complex mechanisms have four-bar l i n k a g e s as elements,

the theory of the four-bar l i n k a g e i s u s e f u l i n d e s i g n i n g of these^mechanisms.

The r e c e n t major c o n t r i b u t i o n i n t h i s f i e l d can be found i n ( 1 ) , ( 25 )

and ( 4 ) . Mechanism dynamics d e a l s w i t h the motion of a mechanism i n response

to a c t u a t i n g f o r c e s , torques and a l s o the f o r c e s and torques produced by a

given mechanism motion. C o n t r o l l i n g f o r c e and torque l e v e l s i s an important

concern i n a v o i d i n g problems of f a t i g u e , v i b r a t i o n and n o i s e . Most of the

present dynamic design procedures s t a r t w i t h a mechanism s k e l e t o n , d i s t r i b u t e

the mass of the members, and add s p r i n g s or dampers to meet dynamic performance

c r i t e r i a a s s o c i a t e d w i t h shaking moment, input torque b a l a n c i n g , and dynamic

time response s y n t h e s i s (25 ) f ( 10 ) . E l a s t i c i t y i n the l i n k s of mechanisms

has a s u b s t a n t i a l e f f e c t on the dynamic behaviour.of the mechanisms. The

5

i n t r o d u c t i o n of c l e a r a n c e s i n mechanisms causes a s u b s t a n t i a l i n c r e a s e i n

the connection f o r c e s r e s u l t i n g i n degradation of l i f e and performance

( 16 ) . I n mechanisms a x i a l loads a r e induced i n the l i n k s by impacts

a t b e a r i n g s u r f a c e s and by o p e r a t i n g loads. Impact-induced a x i a l loads a r e

h i g h l y t r a n s i e n t i n nature ( 1 6 ) . Kineto-elastodynamic a n a l y s i s , which i s

the k i n e m a t i c and dynamic study of mechanisms i n motion i n c l u d i n g the

e f f e c t s o f e l a s t i c i t y and mass d i s t r i b u t i o n has r e c e n t l y been given i n c r e a s e d

a t t e n t i o n ^ 25 ) . Imam and Sandor d i v i d e the complete mechanism design

process i n t o the f o l l o w i n g t h r e e s t e p s :

( 1 ) S e l e c t i o n of the type of mechanism

(2) S e l e c t i o n of the design parameters to s a t i s f y k i n e m a t i c

requirements

( 3 ) S e l e c t i o n of the design parameters to s a t i s f y dynamic and

k i n e t o - e l a s t o d y n a m i c requirements - mass d i s t r i b u t i o n , i n e r t i a

and r e a c t i o n f o r c e s a t the j o i n t s and b e a r i n g s , t r a n s i e n t

and steady s t a t e v i b r a t i o n s , frequency and time response,

e l a s t i c deformation of the components, dynamic s t r e s s i n the

l i n k s , impact, dynamic s t a b i l i t y and b a l a n c i n g .

The b a s i c c r i t e r i a i n d e s i g n i n g and o p t i m i z i n g the a r e a s of c r o s s

s e c t i o n of mechanism l i n k s a r e ( 4 )

1 - The d e v i a t i o n ( e l a s t i c d e f l e c t i o n s ) from the i d e a l performance

( r i g i d body motion) must be w i t h i n the p r e s c r i b e d t o l e r a n c e s

2 - The mass of the l i n k a g e i s to be minimized

4 - Various combinations of the c r i t e r i a such as mass, deformation

and s t r e s s a r e to be minimized

A new method of kineto-elastodynamic design of high-speed mechanisms,

which i s general f o r a l l p l a n a r l i n k a g e s i n c l u d i n g m u l t i - l o o p and multi-degree-

of-freedom mechanisms has r e c e n t l y been presented ( 25 ) . Of s p e c i a l i n t e r e s t

to t h i s work i s (15) which i s considered and d i s c u s s e d i n Chapter 5.

3 The s t r e s s e s i n any of the l i n k s a r e notfbe exceed the endurance 0 ) l i m i t s

6

Chapter 1.

D e s c r i p t i o n of the Machine.

The machine i s a s m a l l v a r i a b l e high-speed ., weaving loom designed

to manufacture ribbon or tape. The main body c o n s i s t s of a r i g i d s t e e l

case e n c l o s i n g the sump and l u b r i c a t i n g mechanism above which i s mounted

_an-H=shaped mechanism u n i t box. The whole complex i s mounted on a

17 - shaped J box s e c t i o n s t e e l platform. The main c o n t r o l box i s mounted

~~on~~the^right s i d e of the body w h i l e the opposing f a c e i s r e s e r v e d f o r the

power t r a n s m i s s i o n . An i n s p e c t i o n cover i s f i t t e d t o the f r o n t of the main

body. The important weaving components a r e mounted e x t e r n a l l y to the r e a r

and top of the mechanism u n i t box. The main body and b a s i c f e a t u r e s a r e

shown i n F i g . (1.1) and P l a t e (1.1)

1.1 Power T r a n s m i s s i o n System

O r i g i n a l l y an " E l e k t r i m 1410 r.p.m., 1.1 kw. 1.5 HP, 415 V. 2.5 amp"

constant speed e l e c t r i c motor was f i t t e d to the r e a r of the main body. The

replacement was a 3 phase, 2 HP v a r i a b l e speed (480-4320 output r.p.m.)

e l e c t r i c motor. The motor was mounted on the r e a r end of the jj -shaped

platform, complete w i t h an a d j u s t a b l e c a r r i a g e which enabled b e l t adjustment

to be c a r r i e d out. AC c u r r e n t i s taken d i r e c t l y from the f l o o r by an i n s u l a t e d

c a b l e fed i n t o the s w i t c h box. A "Fenner B 1800 B69" V - b e l t was used to

tr a n s m i t the power t o the in t e r m e d i a t e s h a f t . F i t t e d to the i n t e r m e d i a t e

s h a f t was a 40 tooth p u l l e y d r i v i n g v i a a timin g b e l t the 20 tooth p u l l e y

f i t t e d t o the main d r i v e s h a f t . The ti m i n g b e l t used was "Fenner 240 L 100"

The V - b e l t a l s o drove the oil-pump s h a f t end d i s c . ( F i g . 1.1). A handwheel/

fl y w h e e l was a t t a c h e d to the main d r i v e s h a f t . The in t e r m e d i a t e V - b e l t

p u l l e y was manufactured to s u i t the a v a i l a b l e b e l t . A l l p u l l e y s were f i t t e d

to t h e i r r e s p e c t i v e s h a f t s u s i n g " t a p e r - l o c k " d r i v e d e v i c e . The two i n t e r ­

mediate p u l l e y s were f i r s t screwed together before f i t t i n g to the s h a f t .

1

CO CD

CD

ifl CO 10 CO CD to

CD CD

CD

o 7 s

to CD CD O 0) o o

CO CD

co

CD

CD •—

CD CD

c9 7? 1/ CD

CO O)

«0

CO

V / CD

CO / CD

CD

CO

/ 1 CD

CD CO CN

CD O CO l±J

II

8

At speeds i n e x c e s s of 2500 r.p.m. (main d r i v e s h a f t speed) s l i p was

det e c t e d between " t a p e r - l o c k " d e v i c e and the main d r i v e s h a f t . Due t o

v e r t i c a l e c c e n t r i c i t y i n the axes o f t h e motor p u l l e y , i n t e r m e d i a t e

s h a f t and main d r i v e s h a f t excess wear was observed on the v - b e l t and on

the t i m i n g b e l t .

1.2 Mechanism U n i t box.

The l e f t ( d r i v e ) and r i g h t (output) end s e c t i o n s of the mechanism u n i t

box, w i t h covers removed, a r e shown i n P l a t e s (1.2) and ( 1 . 3 ) . P l a t e (1.4)

shows the r i g h t end. s e c t i o n complete w i t h connecting rods. I n f i g u r e s (1.2)

and (1.3) the mechanism i s shown as a c t u a l s h a f t s and l i n k s and as p i n

j o i n t e d rods r e s p e c t i v e l y . The main d r i v e s h a f t goes through 0 1 c a u s i n g

the crank arm O^A to r o t a t e w i t h constant angular v e l o c i t y . The r o t a r y

motion o f the crank arm i s t r a n s m i t t e d to the r i g i d t r i a n g u l a r l i n k 4 v i a

the coupler ( l i n k 3 ) . The crank arm;, coupler and r i g i d l i n k 4 re p r e s e n t

an o f f s e t c r a n k - r o c k e r mechanism. The o s c i l l a t o r y motion o f l i n k 4 i s

t r a n s m i t t e d t o s h a f t 0 g v i a the connecting rod l i n k 5 and l i n k 6. The

connecting rods, r i g i d l i n k 4 and l i n k 6 a r e shown i n P l a t e ( 1 . 5 ) . The

s p e c i f i c a t i o n o f the l i n k s a r e given i n the f o l l o w i n g t a b l e :

.Table 1.1

L i n k No.

M a t e r i a l Experimental L o c a t i o n o f Mass c e n t r e s

Experimental Moment of i n e r t i s about mass centre kg - m 2

Experimental BHN

3 Phosphor Bronze a l l o y

12 mm from A 5.84 x 1 0 - 5 117

4 Carbon S t e e l 7. 5 mm jfrom 0_ 30O from "2 C 2

"4.35 x 10~ 5 -

5 Phosphor Bronze a l l o y 1.45 mm from C

-6 7.5 x 10 142

6 Carbon s t e e l 1 mm from 0g .2.26 x 10" 5 -

9

CD

CO

/ CO

CO CD CM

/ c

bi:

CM

03 CO

ID

CO

\

en

c & m »- CO

10

\

P l a t e (1.1) Power t r a n s m i s s i o n system

i

P l a t e i l . 2 Mechanism box l e f t end. From l e f t to r i g h t . Bearings f o r s h a f t s 0_ and 0 0

I I

if)

O \—

"O

"c

c a

c o

"c U J C u

E E E I I <

E E

CN

I I QQ <

E E

CN CN

I I u

CN O cn fN I I CN U CN III CO

O Q O 00 O O

E E

E E o

12

I

"«V i ^ ^ ^ ^ ^ r i v e s h a f t (O.) ' , From l e « to r i ^ - ^ 6

?

•

• i

3 and rods d i n * W i t h connecting

complete 4 Mechani SID P l a t e 1

13

if •

P l a t e 1.5 Connecting rods ( l i n k s 3 and 5 ) , r i g i d t r i a n g u l a r l i n k 4 and l i n k 6

P l a t e 1.6 Main d r i v e s h a f t ( s h a f t 0.)

14

The main d r i v e s h a f t , which was manufactured hollow to allow f o r l u b r i c a t i o n ,

i s shown i n P l a t e ( 1 . 6 ) . The diameter v a r i e s from point t o p o i n t . A

d e t a i l e d crank-arm end drawing i s shown i n F i g . ( 1 . 4 ) . The r i g i d t r i a n g u l a r

l i n k 4 i s attached t o s h a f t 0 by a clamp mechanism and secured w i t h a key,

w h i l e l i n k 6 i s a t t a c h e d by a s i m i l a r clamp mechanism but without a key.

At high crank arm speeds (approaching t o 2500 r.p.m.) s l i p between l i n k 6

and s h a f t 0^ caused stoppages and mechanical damage. L u b r i c a t i o n was

hydrodynamic i n the s h a f t b e a r i n g s . L u b r i c a n t e n t e r s the bearings v i a

converging channels t o support the s h a f t s without s h a f t t o b e a r i n g c o n t a c t .

Hollow head cap screws were used to f i t the b e a r i n g s to the frame. The

s h a f t s were not secured a g a i n s t movement i n a d i r e c t i o n p e r p e n d i c u l a r to

t h e plane of the mechanism. At r i g h t angles to the a x i s o f the s h a f t 0

i s the r o t a t i o n of the needle arms produced by p a i r s o f b e v e l gears on the

needle arm base and s h a f t 0 .

D e t a i l e d drawings of l i n k 6 and s h a f t 0 g w i t h the comb, a r e shown i n

F i g . (1.5) and ( 1 . 6 ) . The h o r i z o n t a l p o s i t i o n i n g : of the s h a f t s a r e shown

i n F i g . (1.7) The s p e c i f i c a t i o n of s h a f t s 0 , 0 - and 0 a r e presented i n

Table 1.2. Table 1.2 ,

sha f t no.

mass kg.

Exp. moment of i n e r t i a about m a s s 0 c e n t r e s

kg^nr Fu n c t i o n

°1 1.409 8.72 x l o " 4

( w i t h the hand wheel) Main d r i v e s h a f t d e l i v e r s power

to the system

°2 0.699 9.22 x 1 0 ~ 5

( w i t h the bevel g e a r s ) C a r r i e s a p a i r of p a r t i a l bevel gears to r o t a t e the needle arms

°3 0.805 3.06 x 1 0 - 4

( w i t h the combs) C a r r i e s a p a i r of combs f o r weaving

15

«r—«•

o I — L L

if)

CD CNI

CNI bo

04

•a •a (D to

CNI CD CN a)

i T3 0) CO

I

> I I

(0 CN V)

0

bo

LO CO LO

16

o

n O)

in to

a) n CO

II T 3

I

- t

o LO

ftf

-H en K

1 r n -a

17

bO

o *— II

CO I

CO

7

\ t o

r <0

\ 0

CD CO

i i LO

LI

18

CO

tr

LU 1

ID LD

I

1

CO OQ

ID

en

Q C3 EC

LL U J

o \ CO

M l

C3 \ COl

\

Vd

\

NI n ( J 17}

he Cfl

UJ UJ > cc Q

19

The experimental moment of i n e r t i a values presented i n Tables 1.1

and 1.2 are determined by a t r i f i l a r suspension. The needle arms and

shaft Og with the comb and bearings are shown in Plates 1..9 and 1.10.

The masses of l i n k s 3, 4, 5 and 6 are: 0.116 kg, 0.116 kg, 0.0283 kg and

0.0749 kg respectively. The mass centres of the l i n k s are determined by

using a knife edge. The top view of the mechanism unit box and shaft 0

are shown in plates 1.7 and 1.8 respectively.

20

• I i

Mechanism u n i t box. Bevel gears c a r r i e d by l i n k 0 P l a t e 1.7 Top view weft needle arm, s h a f t 0_ and the reed

P l a t e 1.8 Shaft 0 s h a f t bearings, a nd the bevel gears

21

V

P l a t e 1.9 Needle arms

is

P l a t e 1.10 Shaft 0 comb and bearings

22

Chapter 2

Kinematic Analysis of the Mechanism

2.1. Introduction to Kinematic Analysis of the Mechanism

Recently a variety of methods have been developed for dynamic and

kinematic analysis of mechanisms such as methods based on kinematic

constraints, motor'a'lgebra, matrix methods, quaternion and dual-number

methods, r e l a t i v e motion and incremental equations u t i l i z i n g numerical

methods, i n addition to the c l a s s i c a l methods. Other basic methods are;

Quinn's energy d i s t r i b u t i o n method, Lagrangian method, solutions employing

complex polar notation and complex numbers, kineto-elastodynamic analysis,

Raven's analysis, velocity analysis by instantaneous centres, velocity

analysis by components, velocity and acceleration image method, graphical

and a n a l y t i c a l velocity and acceleration analysis and various algebraic

methods. In the following sections a complete kinematic analysis of the

mechanism i s presented to specify the motion of the mechanism and to

determine the kinematic values. I t i s assumed that a l l the l i n k s of the

mechanism move in the same plane and the crank speed i s constant.

2.2 Geometry of the Mechanism

From F i g . (2.1); for . 0° ^360°

C L E cos( f.) (0.0.11)

AE s i n ( « ) (0.0JL1)

AF AE + 0.005

0.048 - 0,E

AO (AF) + (0„F) 2

AF tan -1

23

CM

(JJO •H p4

6 Z 0-0

24

-1 0.00128 - ( A 0 o ) 2

zek = cos 2

(-0.044) ( A 0 2 ) 2

-1 0.002248 - ( A 0 „ ) 2

c a l = cos 2

(0.001848

<x = 180° - (zek + c a l )

B = a - Tl

Y = 180° - ( 71+ zek + 75° )

_ ! 0.032 - s i n ( y ) (0.022) 9 = tan

0.042 - cos (y) (0.022)

0.032 - s i n (y ) (0.022) CO 3

van == cos

si n ( fi .)

-1 ( 0 . 0 2 9 ) 2 + ( C 0 3 ) 2 - ( 0 . 0 2 ) 2

( 0 . 0 2 9 ) 2 (co ) 3

x \ = van + fl

-1 ( 0 . 0 2 ) 2 + ( C 0 o ) 2 - ( 0 . 0 2 9 ) 2

CO = cos 3 (0 .04) ( C 0 3 )

upa = c o - teta

yum = 180° -(x^ + upa)

a l l the angles are i n degrees. Relationships between ^ , y> x X a m * u P a

depend upon the lin k lengths. Numerical variation of y, Q, x\ and upa with

^ are given i n (A 1 ) and are shown graphically i n Figures ( 2 . 2 ) , (2*3)

(2 .4 ) and ( 2 . 5 ) . Variation of the transmission angles, c a l and yum with

^ are shown in F i g . (2 .6)

2 .3 Velocity and Acceleration Analysis - Analytic method

Angular positions of l i n k s 3 ,4 ,5 and 6 are determined r e l a t i v e to the

x axis

25

in CL 0

0

0

CD

i.0

in as CD

in 0 bO

/ /

in / 0)

\ bo as

CO CD

0

cs

bo 0 bo

as O CD

\ \ as

as

CM / LO LO

bO

28

30

The velocity of A i s ;

VA = OjA'.tug q (where 9 shows angular position)

where 9 = j, + 90°

(Bg = angular velocity of the crank arm i n rad/s

The velocity of B i s :

where 9 = y + 165°

The veloci t y of B with respect of A i s

V- = V b a ^ 9

where 9 = g + 90°

the directions are assumed for V and V . may be incorrect. Subsequent

calculations w i l l indicate whether t h i s i s true or not.

The. r e l a t i v e velocity equation i s ;

V = V. + V A B A BA

transforming V^, and into complex rectangular notation;

+ j y B ) = <* A + j y A ) + ( * M + j y M ) (2.1) c o

where and y^ are known

K_nowing the directions of V and V ; B BA

y B 0

TF- = t a n ( Y + 165 ) (2.2) *B

y£A = tan (g + 90°) (2.3) *BA

* • »

solving equations ( 2 . 1 ) , (2.2) and (2.3) simultaneously, x ^ , y^, x Q and

can be determined.

The angular velocity of l i n k s 3 and 4 may be calculated as;

v v ® 3 = _BA and u>4 = B AB 0 oB

31

from the geometry of the mechanism;

v = v ( i n magnitude) B C

C C

where

e = y • K 9 0 °

the velocity of D i s ;

V = V ^ a D D where

6 = 90° - u p a

the velocity of D with respect of C i s ;

V v A DC DC 0

where

0 = x x + 90° the r e l a t i v e velocity equation i s ;

% " VC + V transforming into complex rectangular notation;

(*D + JV = (*C + J' yC ) + (*DC + j y D C ) ( 2 ' 4 )

knowing the directions of and V

yD = tan (90 - upa) (2.5)

XD

y JXL = tan (x\ + 90°) (2.6) x DC

solving equations (2.4),(2.5) and (2.6) simultaneously, 0 0 # V •v v v and y„ c a n be determined DC' yDC' D D

thus, the angular velocity of l i n k s 5 and 6 are;

32

U) = DC and OJ = ^ o o — DC DO o

the components of acceleration can be obtained as follows;

A A * = °1 A *2 * 0

where o

e = t + 90

A A* = 0, since = 0

V = °1 A- ">22 8

where

9 = -(180° - .* )

ABA r = A B - ^ 9

where

9 = -(180° " 3)

A B r = °2 B' » 4 2 Z - 0

where

9 = "[180° - (75° + Y)1

from the geometry of the mechanism;

r r A = A_ (i n magnitude)

A * = A * ( i n magnitude) B L

t t both A A and A„ are unknown and are to be dete'raiined BA B The r e l a t i v e acceleration equation i s ;

A * + A nr = A * + A A

r + A * + A „ Ar (2.7) B B A A BA BA

transforming the acceleration components into complex rectangular notation

and substituting into equation (2.7) produces

33

t , t <x B + dy B ) + ( x Br + j y B

r ) = ( x Ar + j y / ) + ( x ^ + J y ^ * ) + + j y / )

(2.8)

the required slopes of these components are such that;

" t = tan (v + 1 6 5°) < 2- 9>

V r X B

t V n A „ o -p- = tan (90 + p) (2.io) "flA

by equating the re a l and imaginary parts of equation (2.8) and solving t ** t " t " t simultaneously with equations (2.9) and (2.10) gives x , y , x and y_,A B Q Bn BA

The acceleration of B i s ;

A n = A * + A r

B B B

where . t " t V t A„ = x„ + j y B B J , B

acceleration of l i n k 3 i s ;

ABA = AW + ABk where • •

A t t t BA = XBA + j yBA

the angular acceleration of l i n k s 3 and 4 are;

a t a t Of = BA 01 = B

AB 0 2B

si m i l a r l y , the r e l a t i v e acceleration equation for l i n k s 4, 5 and 6 can be

written as

A * + A * = A „ r + A * + A * + A _ r

D D C C DC DC (2.11)

34

where

A Dr = D0 3 u ) 6

2 Z e ; e = 360° - upa

A D Cr = DC u, 5

2 Z e ; 0 = 180° + xX

cr = a

cr Z 9 ; 9 = 1 8 Q ° + y

t t , o c = a c

A~ = a- Z 9 9 = - + 9 0

x c and y c are therefore known as well as x c and y c . Again by transforming

equation (2.11) into complex rectangular notation; " t ** t " r " r r " T " t *" t J° t ** t

< XD + J yD } + U D + j y D r ) = < X C r + J yC

r ) + ( X C + ™C > + ( XDC + J yDC > + ^ + ^ D C ^ ( 2 - 1 2 )

the required slopes are;

y x

J l _ = tan (90° - upa) (2.13) 'XD

and

•• t y

= t a n ( x x + goO) ( 2 1 4 ) XDC

t *" t " t "* t from equations ( 2 . 1 2 ) , (2.13) and (2.14), X

D » y D » a«d C J» n b e

obtained. • o

A t t , . t AD = ^ + j yD

the acceleration of D i s ; A = A * + A r

D D D angular acceleration of l i n k s 5 and 6 are;

t

5 DC t

aD D 0 3

50 35

US G3

\ CO

J

1

73 t I \

(3

\

\ \

Q. \

CM

bp

/ /

/

O CD

CD

S

CQL_

36

Variation of the angular acceleration of l i n k s 3,4,5 and 6 with crank angle i|f are shown in F i g . (2.8). F i g . (2.9), F i g . (2.10) and in F i g . (2.11). Variation of angular v e l o c i t i e s of l i n k s 3,4,5 and 6 with the crank angle ty. are given i n Fig. (2.12) for a crank speed of 2500 r.p.m. Ratio of angular velocity of the individual l i n k s to crank arm angular velocity i s shown in F i g . (2.13), for a complete revolution of the crank arm. A l l the kinematic values calculated above, are presented i n (Al)

for a crank speed of 2500 r.p.m.

2.4 Algebraic Approach

The system i s separated into two loops, Fig. (2.14, a and b) F i g . (2.15)

The f i r s t loop represents a crank-rocker mechanism:, also known as a crank-

lever mechanism, which i s a popular type of the well known four-bar linkage

for converting continuous rotary motion to o s c i l l a t i o n . The second loop i s a

four-bar chain attached to the r i g i d l i n k 4 of the f i r s t loop at point C.

The two extreme positions of the output lever ( l i n k 4) of the f i r s t loop are

expected to occur when crank arm ( l i n k 2) i s in l i n e with the coupler. i

Proportions of the mechanism s a t i s f y Grashoff's c r i t e r i o n as in the following:

Base l i n k ^coupler + (output lever-driving crank)

4.8 <" 5.3

Base l i n k ^ coupler - (output lever - driving crank)

4.8> 3.1

Let A,B,C and D be lengths of input crank, coupler l i n k , output lever, and

base l i n k of the f i r s t loop, F i g . (2.14a) O , ff2 = a n S l e s showing dead-centre positions of output l i n k

Aff = o>2 - m1

A 6 = 0 2 ~ '01 ~ TT

the unit of angles being i n degrees.

CO. U

S-l CO

38

Q_ O I CO

•H in a

a> r-i ho c co

a c

H o

a>

•p .

s: p •H

TP

> c • H

•H P i-i •H in o o a

* r bO b c

•H c o

•H .P m •TO CO

a) c .>-< (0

o CO o p co

U) •i •H

CD i - l a> 3 U) bO -6 G CO

o ! H o o 1-1 o c o u •H <D

. P p CO c

•H 3 * h o

CO o

> Oi CN

h i •H

I™ CM 3

I'M

XJ

X \ \

o CM

\

\ o \

\ \ \

o in

/

/ /

o

/ , o

r -t o

o CM O

CN O I

I D

O I

39

40

in

CD

ft t-JO

CO

G)

ft

CD

10 eg 0)

cq CO p

in

TO CD

0) CO UP)

CO 10

(!) so - H CO

CD Cu

CD

CM

t>0

evi CSI

tn

[/)

u O u CD -•—> C D O u

0» n

a •4—»

If) CL If) a: c o o > O — LD u CNJ

'<7i II v_ o a v . Q.

U) t, J£

C jf a V—

u

CO LU > tr ZD u

42

so TO

CN 5

CM CN 5 $

LO CD

\ \

\ o CO CO X CN X CO

X /X. ; x X /

X o \ o X CM X

X X /

CN / \ O i

/ CD / /

o

X o X CD \ / o o CD LD

o o o o o

43

By using complex algebraic notation, the displacement equations can be

expressed as;

i fl. ±oij (A + B) e = D + C e (2.15)

iB2 i(02 " TT) iog Ae + Be = D + Ce (2.16)

equating r e a l and imaginary parts of equations (2.15) and (2.16) respectively;

(A + B) cosg 1 - C cosc^ = D (2.17)

(A + B) s i n p x - C sinofj = 0 (2.18)

(A - B) cos0 2 - C cosa 2 = D (2.19)

(A - B) s i n 0 2 - C sinor 2 = 0 (2.20)

from equations (2.17) and (2.18);

C sine?. D + C COSQT-A + B = = - (2.21)

s i n COSQJ.

from equations (2.19) arid (2.20)

C sino/g D + C coscv„ A - B = = (2.22)

sinf3 2 c o s B 2

from equation (2.21)

C 5 i n 9 l

sxn (a1 " B x)

from equation (2.22)

C sing

D s i n ( a 2 - p 2)

equations (2.23) and (2.24) are combined to give;

(2.23)

(2.24)

44

CO

CM CM

CM ho

bO /

.1

/

8"

\ CN o

CD

/ / I

CD

o

_J

45

ID

01

in

in

•j)

0)

3)

CM W)

\

I I I

in O

«T3 i - cn

46

s i n 0 s i n - 0 X)

s i n ( e i + A s ) s i n (Q^ + & Q R - ^ - A B ) (2.25)

by expanding equation (2.25) and making the following substitutions;

Y = ffj + Ac? - AP

6 = Q f l

X = AS

p = cos 5 cos \ - cos Y

q = s i n y + cos 6 s i n \ - s i n 6 cos \

r = - s i n g s i n \ 2

p tan q tan + r = 0

where x = tan 3^

y i e l d the quadratic equation;

2

p x + qx •+ r = 0

where the i n i t i a l angle of the input crank i s ;

(2.26)

(2.27)

3, arc tan (+ x)

(+ x) being the two roots.

There e x i s t s only one r e a l root which gives the solution of Sj

i n i t i a l angle of the output lever arm can be calculated from;

The

<*1 arc tan a (k + b)

bf - ah (2.28)

where

Z = -0 X + Aa - A0

a' = s i n

b = s i n 0r

47

f = cos 0 h = cos ft

k c s i n 5

from equations (2.15) to (2.28), the following values have been calculated

for loop 1;

P 2 - 2 0 1 °

3, = 25°

ft o

«2 " 1 5 1

I t i s desirable to have the transmission angle d e v i a t i o n a s small as possibl

throughout the range of operation. The minimum permissible transmission

angle depends on the magnitude of the transmitted forces, j o i n t f r i c t i o n

and manufacturing tolerances. The extreme values of the transmission angle

are shown in F i g . (2.14 b). Prom .the c o s i n e law;

u, . = arc cos mm B 2 + C 2 - (D - A ) 2

2 BC (2.29)

= arc cos max

B 2 + C 2 - (D + A ) 2

2 BC

and .2 2 2 2 A + D = B + C

while;

C = i i n P l sxn

D or C = 02

(2.30)

48

A =

where;

s i n a. s i n au r = 1 r = ^ 1 2 s i n 0 X s i n p g

s 2 = s i n (» 2 - p 2)

substituting numerical values into equations (2.29) and (2.30)

u . = 62° nnin

(J-min — 1 3 4 °

Ap, ^ p, - p, , =72° max min

In order to have a smooth motion throughout the whole range of operation,

transmission angle i s o f t h e utmost i m p o r t a n c e . F r e u ' d e n s t e i n (1 ) n a s

shown that a good choice of transmission angle also coincides with minimum

^ overtones of the output lever, although a large transmission angle does

not n e c e s s a r i l y guarantee low fluctuation of torques. The force transmission

from the coupler to the output lever i s i d e a l l y e f f e c t i v e when the transmission o o angle i s nearly 90 or deviates as l i t t l e as possible from 90 . A good

discussion of the equations and procedure applied above are given in ( 1 ) .

Dead-centre positions of l i n k 6 and l i n k 4 are shown in F i g . (2.15). Variation

of the transmission angle, yum, with crank angle p s i has been presented i n

F i g . (2.6). O s c i l l a t i n g motion exercised by l i n k 4 i s transferred to l i n k

6 via the intermediate l i n k 5.

2.5 Application of Raven's Analysis

As shown i n F i g . (2.16) each l i n k i s replaced by i t s position vector.

Two seperate reference frames have been taken, x-y and x^-y^ corresponding

to two seperate but dependent loops, loop 1 and loop :2, writing the summation

law for the f i r s t loop, gives

49

+

CO

CM

CO CD

\ \

or r CO v y CD y \ -7" CD

a:

I CD CM CD

CD

co — I or

or

00 CP

CD

50

R + R 2 + R g + R 4 = 0 . (2.31)

Transforming to complex notation,

5 + r 2 e + r 3 e

In d i f f e r e n t i a t i n g eq. (3.31a)

r^e + r 2 e + r 3 e + r 4 e = 0 (2.31a)

j r 2 0 2 e + jr 3Gge + j r ^ e = 0 (2.32)

r^» r2» r3» r4 and 0^ are constants. • » 0

Letting 9 2 = uig , 9g = u)g and 9 4 = u>4 , gives

19 2 J 63 J e 4

j r 2 u ) 2 e + jr 3U)ge + j r ^ e = 0 (2.32a)

Equation (2.32a) contains the following quantities;

VA = r2u^' VBA = '3*3' VB = W

and i s the solution of the equation

V B = VA + V

After a transformation to complex rectangular notation and a separation of

the r e a l and imaginary terms eq. (2.32a) becomes;

r2 t u2 c o s 0 2 + X3W3 c o s Q 3 + T4W± c o s 9 4 = 0 ( 2.33a)

2 ® 2 s i n 0 2 - r 3 u ) 3 sin0g - r ^ s i n 9 4 = O (2.33b) - r,

The unknown quantities i n equations (2.33a) and 2.33b) are^g and ^ .

Since there are two equations and two unknowns, UJ„ and m. can be determined.

By d i f f e r e n t i a t i n g equation (2.32a), using a uniform angular velocity

of 2, gives

.2 • . 2 . . J 3 .2 « J 3 . « J 4 .2 • 4 0

J r2a>292° + J ^ S 6 + j r 3 ° 3 % e + + 3 W4 9 = °

(2.34)

51

2 . • = Q n where j = -1 and ^ = Qf

2 j e 2 j e 3 2 j e 3 ^ ^ - TgUJg e + j r <* e - r ^ e + r ^ e - r4<o4 e = 0 (2.34a)

the terms a r e i d e n t i f i e d a s ;

r 2 a a = r2<«2

r 2 3 B A = r3 U J3

r 2 A B ~ r4t«4

A B A T = r3,«3

V = r4 f f4

a * = 0 a

equation 2.34a corresponds to the vector equation,

AT.* + A „ R = A.* + A , R + A,,.* + A.r. . B B A A BA BA

corresponding t o t h e a c c e l e r a t i o n p o l y g o n .

using the equation

J 9 e = COSQ + j s i n 9

equation (2.34a) can be transformed into complex rectangular notation.

Separating the re a l and imaginary terms gives;

- r 2 u ) 22 cose 2 - r 3cv 3 sinOg - r ^ 2 cos0 3 - s i n 9 4 r ^ 2 cos0 4 = 0

(2.35a)

2 2 2 " r2Sfe S i n 9 2 + r 3 a 3 c o s 9 3 ~ r3{D3 s i n 0 3 + r 4 a 4 c o s 8 4 " r ^ s i n 9 4 = 0

(2.35b)

52

the only unknowns i n equations (2.35a) and (2.35b) are Q?3 and Q >4. The

simultaneous solution of these equations u t i l i s i n g determinants y i e l d

values of a and o^-

Writing the summation law for the second loop gives,

„ 1 1 1 1 R l + R 2 + R 3 + R4 = °

Transforming to complex notation 1 1 1 1

x J Q 1 x J Q 2 J6"3 J9.4 r, e + r. e + r_ e + r . e = 0 (2.36) 1 2 3 4

1 1 1 1 ^ 1 ^ where , , , r^ and 9^ are constant

d i f f e r e n t i a t i n g equation (2.36) gives 1 1 1

J 9 2 J 0 3 J 94 j r 2 0 2 e + j r 3 °Q3 e + e = 0 (2.37)

0 1 Letting 0 U ) 4

n 1

6 3 = w5

' 1 4 6

J82 1 J03 1 j e 4 X

J ' 2l u ) 4 e + j : r 3 l u ) 5 e + J r 4 l t B 6 e = 0 (2.37a)

equation (2.37a) contains the following quantities;

V c = r 2 ^ 4

VDC =

VD = U \

and i s the solution of the equation

V = V + V D C DC

53

After a transformation to complex rectangular notation and a seperation of

the r e a l and imaginary terms eq. (2.37a) becomes;

1 1 1 1 1 1 / n 0 0 . R

2 U ) 4c o s e 2

+ R3 U ) 5 C O S e 3

+ R4 « M 6

c o s e 4 = 0 (2.38a)

- r2

1 ( B 4 s i n 9 2 1 ~ r 3 1 ° l > 5 s l n e 3 1 ~ r 4 l u ) 6 S i n B 4 1 = ° (2.38b)

The unknown quantities in equations (2.38a) and (2.38b) are (u_ and u)e • 5 o

Solving them simultaneously gives o)§ ^nd (0g- D i f f e r e n t i a t i n g equation

(2.37) gives;

1 1 1 1 1 J 02 ^ 2 ^ 3 ^ 3

. 1Q 1 1«1 2 . l " 1 1«1 2 . 1'* 1 J r 2 62 6 " r2 9 2 6 + J r 3 9 3 6 " r3 9 3 6 + J I 4 94 G

1

where i : i 2 0 2 = ff4 9 2 = (n4

B 1 a fl1 2 - in 2 93: = 5 9 3 " '"s

e 4 i = « 6 01 42 = ^ 2

, 1 . 1 . 1 . 1 .1 . 1 J 9 2 1 2 J02 , 1 J0.3 1 2 J 9 3 . 1 a

J f i 4 J r 2 a 4 6 " r2 ^ 6 + J r 3 a 5 e • r 3 ""S 6 + J r 4 ff6°

1

- V ^ 4 = ° < 2 - 4 0 )

the terms are i d e n t i f i e d as;

r 1 2 a c ~ r2 «4

t 1 a c = r2 «4

r 1 2 — r

DC 3 «5

54

t 1 Dc " r 3 a5 r 1 2

aD = r4 <»6 t 1

3D = r4 a6

equation (2.40) corresponds to the vector equation,

A * + A r = A * + A „ r + A * + A * D D C C DC DC

corresponding to the acceleration polygon using the equation

e J9 = cos 0 + j s i n 9

equation (2.40) can be transformed into complex rectangular notation.

Seperating the r e a l and imaginary terms gives;

- r 2 a 4 s i n 9 2 - r 2 ^ cos 9 2 - r g ^ s i n 9 3 - r 3 ^ cos 9 3

" S i n 9 4 X ~ T<k®G C O S ^ = ° (2.41a)

1 1 1 2 . 1 1.. 1 1 2 . . 1 r 2 cv4 cos e 2 - r 2 ^ s m 9 2 + r g a g cos g 3 - r g ^ s i n 9 3

+ ' ^ f f e - 0 0 - 8 9 4 X " r 4 1 ( U 6 2 S i n 9 4 X = 0 (2.41b)

The only unknowns in equations (2.41a) and (2.41b) are a 5 and a^. A

simultaneous solution of these equations y i e l d values of ov and oia.

2.6 Graphical Approach

The velocity and acceleration polygons have been solved graphically

for a set of different crank positions and are presented in(A2).

55

2 . 7 Energy variation of the system

The t o t a l k i n e t i c energy of the system at any instant i s composed of

the k i n e t i c energy of translation about the mass centres of the li n k s and

the k i n e t i c energy of rotation about the mass centres of the l i n k s and

shafts. This can be expressed as i n the following form;

2 2 2 2 2 2 2 Total k i n e t i c energy = \ T I Q J U ) + I 3u> 3 + I 0 2 u 5 4 + I4t"4 + I5^5 + I6 U )6 + I 0 3 U J 6 ^ 1

2 2 2 2 2 + 4 Tm-V _ + m_V _ + m.V . + m_V + m„V 1 ( 2 AO\ 2 2 g2 3 g3 4 g4 5 g5 6 g6 1 K*.**)

where V , ' = l i n e a r v e l o c i t i e s of the mass centres of the corresponding gl-»6 l i n k s

2 2

in equation ( 2 . 4 2 ) IQ^OVJ A N ° " M 2 VG 2 a r e c o n s t a n t s (since cv = 0 ) . The var­

i a t i o n of the k i n e t i c energies due to rotation of shafts 0^, 0^, l i n k 4 and

link 3 with the crank angle as shown in F i g . 2 . 1 7 . The t o t a l k i n e t i c energy

due to translation i s negligible i n comparison with the k i n e t i c energy due

to rotation. Therefore equation 2 . 4 2 can be reduced to;

2 2 2 2 2 2 Total k i n e t i c energy? K I

3 0 ) 3 +- , I

0 2«>4 + I4")4 + I5«»5 + I6 U )6 + I03 U )6 ^ + C

where C = \ I

( 2 . 4 3 )

2

01«2

variation of the t o t a l k i n e t i c energy (excluding C) with the crank angle i s

shown in F i g . 2 . 1 8

56

B CO V)

M 9

CM co i «-> CO

0 e m to

e c u O CB

ro CS CO c c ( Y°

«

ro

\ / I

i

\ CNJ

e n

c 4 6)

if) CM CM

5 7

F i g . 2.18 V a r i a t i o n of the k i n e t i c energy w i t h the crank angle

/

\

1 crank speed= 2500rp. A=3.712

1 I

\ \ \ / / \ 1/

pS| i i 300" 360° 90° 210'

58

Chapter 3 Force and S t r e s s A n a l y s i s 3.1 Kinetostatic Approach

The motion of the mechanism i s completely specified and the. purpose

i s to compute the bearing reactions, shaking forces, shaking moments as

well as the forces and the torque required to produce the motion. The

s i z e , shape and material of each l i n k are known. The i n e r t i a forces

are taken as i f they acted at a point on the l i n k s , although they are

distributed along the l i n k s and are not concentrated at one point. In

the following analysis the concern i s only with the forces at points

where the l i n k s are paired with other l i n k s . These forces are treated

as external forces on the l i n k s . The shapes of the l i n k s are assumed to

be r i g i d and the pin j o i n t s are considered to be f r i c t i o n l e s s . The

i n e r t i a e f f e c t s of the individual l i n k s , in comparison with the i n e r t i a

e f f e c t s of the shafts through 0 and 0 are reasonably negligible and

t h i s aspect of the dynamic analysis has been shown i n further steps.

However for a detailed dynamic analysis of the system, which i s beyond the

scope of the objective, the d i s t r i b u t i o n of the i n e r t i a forces along the

l i n k can be important. In most advanced engineering cases the stresses

due to i n e r t i a forces are determined by breaking the link into equal

length sections. The i n e r t i a force for each section i s determined from

the mass of the section and the acceleration of the midpoint of the

section which represents the d i s t r i b u t i o n of the i n e r t i a forces along the

l i n k (21 ) . In cases where the l i n k s are not of uniform cross section

the accuracy of t h i s approximation depends upon the number of sections

that the l i n k i s broken into. By increasing the number of sections a

greater accuracy can be achieved.

59

A solution to the dynamic force analysis of a four-bar planar

mechanism has recently been presented ( 4 ) , as a set of algebraic

equations. The forces along the li n k s 3 and 5 cause pure normal

stresses, and the normal forces cause bending st r e s s e s . In the actual

case the normal stress e s vary along the l i n k s because of the i n e r t i a

forces. The system has a single degree of freedom, the angular pos­

i t i o n s of l i n k s 3, 4, 5 and 6, given by 0 g ( t ) =3, 0 4 ( t ) = y + 75°

0g ( t ) = x\ , and 0g ( t ) = 360° - upa, are functions of the angular

position 0 2 ( t ) = ty., and the length of the l i n k s . The lin k lengths

are denoted by 1^ , i = 2, 3, 4, 5 and 6, and each of the moving l i n k s

has mass , i = 2, 3, 4, 5, 6 and a moment of i n e r t i a 1^ with respect

to the centre of mass. The locations of the centres of mass of the

members are defined by parameters 1^ and 0^. The bearing reaction F ^ j

i s the force of member i on member j . The D'Alembert couple on lin k i i s ;

C 4 = r I . 0. i 11

The x and y components of the D'Alembert force being;

ix m. aj l i x

and

m. a. i i y

where a ^ and a ^ are the corresponding acceleration components of the

centre of mass of member i . The motion of the mechanism i s known and the

i n e r t i a loading on the system i s defined. T i s the external torque on

the input l i n k 2, required to produce the prescribed motion. The dynamic

60

equilibrium equations for the fi v e moving l i n k s y i e l d the following

system of f i f t e e n l i n e a r algebraic equations:

where

ti

where

D2x + F12x " F23x = ° ( 3 - 1 }

D2y + F12y " F23y = 0 < 3 2 > D3x + F23x " F34x = ° ( 3 ' 3 )

D 3 y + F23y - F34y = ° ^ ( 3 " 4 )

4x 34x 14x 45x

% + F34y + F14y " F45y = ° ( 3 " 6 )

D5x + F45x " F56x = ° ( 3 ' 7 )

D. + F - F = 0 (3.8) 5y 45y 56y

D + F - F = 0 (3.9) 6x 56x 67x D + F - F = 0 (3.10) 6y 56y 67y

C 2 = " h *2 C 2 = -hlk

C 2 = " ( I 2 + m 2 r 2 + V \

*01 = M o m e n t °* i n e i " t i a °f the driving shaft through 0^

C - D r s i n 0 + D r cos 0 + T + F 1 s i n 0 - F 1 cos0' 2 2x r2 v2 2y 2 ^2 2 23x 2 ^2 23y 2 2

op C = - 1 0 3 3*3

C 3 " D 3 Xr 3 S l n 0 3 + D 3 y r 3 C ° S 0 3 + F 3 4 x 1

3S i n 0 3 " F 3 4 y 1

3C ° S 0 3 = °

c4 = - 1 A C' = - I 0 4 02 4

I „ = Moment of i n e r t i a of the shaft through 0 Oa 2

C4 = - ( I 4 + V 4 + I O 2 ) 0 4

61

where

C \ ' " D / l r y l S l n ^ / l + 9/1 > + D / l T A C O S (0/1 + 9, > 4 4x 4 4 4 4y 4 4 4

- F_. l . s i n 0. + F_. 1. cos 0. - F.._ 1. s i n 0* 34x 4 4 34y 4 4 45x 4 4 + F,. 1, cos 0! = 0 (3.13)

45y 4 4 C = - I 0

5 5*5

C 5 " D 5 x r 5 S i n 0 5 + D 5 y r 5 C O S 0 5 + Ws81" 0 5

-F56yhC°S05 = ° (3-14>

C 6 = -h*6

°6 =

I Q g = Moment of i n e r t i a of the shaft through 0^

C 6 = " ( I 6 + m 6 r 6 2 + I 0 3 ) 0 6

C 6 " D 6 x r 6 S i n < ' 0 6 " D 6 y r 6 C O S 0 6 " F 6 5 x L 6 S i n 0 6

+ F 6 5 y 1 6 C ° S 0 6 = °

(dot denotes d i f f e r e n t i a t i o n with respect to time t )

The configuration of the mechanism and the free-body diagrams are

shown i n Fig.(3.1-3.2) f o r 0 g ( t ) = . ty. = 50°. Relations of the form

F ^ ^ = ~ F34x' e t c , » n a v e D e e n employed in the formulation of these

equations. The shaking force F g J i s the resultant force on the frame.

The x and y components of the shaking force are;

F = F + F + F = -F - F - F sx 21x 41x 67x 12x 14x 76x

= D2x + D3x + °4x + D5x + D6x ( 3 " 1 6 )

and

F = F + F + F =-F - F - F = sy 21y + r41y 67y *12y *14y r76y

D_ + D 0 + D + D_ + D c (3.17) 2y 3y 4y 5y 6y

62

o

(D

\

\ LO

CN -O CD CD •-J

\ \

ro \ \ \ \

rxJ ro

01 CM CN

64

The shaking moment M about an a r b i t r a r y point 'p' on the frame Fig. sp

i s

M s P = " T2 " F 4 1 x e i S l n *1 + F 4 1 y e i C O S h

- F21x e2 S i n *2 + F21y e2 C O S *2 " F67x 63 S ± n *I

+ F 6 7 y e 3 C O S *3 ( 3 1 8 )

The effect of gravitational p u l l on the l i n k s has been neglected.

The compressive and/or t e n s i l e forces ac.ting on the l i n k s can be

determined by resolving the forces acting on the l i n k s along themselves.

65

3.2 - Application of V i r t u a l Work Method to the Mechanism

Energy methods can be used to short-cut the previous kine t o s t a t i c

approach. The following solution, u t i l i s i n g the method of v i r t u a l work

introduces greatest timesaving to the analys i s . The main advantage of the

method i s that, i t eliminates the l i n k - t o - l i n k treatment, and permits

an examination of the whole system at one time. A good discussion of

thi s method i s given in (8), and ( 9 ) .

For any mechanism composed of n members, the method of v i r t u a l work i s

written:

T . 0- + F .V + (-m A '.V ) n n n n n Gn Gn

• < - v „ ^ > = ° < 3 - w )

which can normally be solved for one quantity. Since the terms are vector

quantities, the solution includes both the magnitude and direction of the

unknown. I t s major disadvantage i s that s i n c e eq 3.19 c o n t a i n s only

the a p p l i e d f o r c e s and torques, i t can not be used to solve for internal

forces or the reactions between members of the mechanism.. Formulation of

eq. 3.19 u t i l i z e s an imaginary small displacement of the mechanism, being

consistent with the constraints of the mechanism. The work done by the

v i r t u a l displacements i s referred to as v i r t u a l work and i f the system

i s in s t a t i c equilibrium under the action of the applied forces and

torques then the work done with a v i r t u a l displacement i s zero.

Application of eq. 3.19 to the mechanism yie l d s .

66

T 2 °2 + ( " m 2 a 2 - V 2 ) + ( " m 3 V V + < _"4 V V + ^sVV + (- m6 V V + ( _ I 3 a3 °3 } + + [ _ ( ( I 4 + m 4 r 4 2 ) <*4 ^

+ ^ ( ( I 6 + m6 r 6 2 ) + I 0 3 ) *6 = ° ( 3 ' 2 0 )

F V = 0 n n

Substituting the appropriate values for 'ijr' equation (3.20) i s

used to calculate the external torque T . In the formulation of equation

(3.19) the gravitational effects are neglected. A modified form of t h i s

equation taking account of gravitational e f f e c t s i s given i n ( 8 )

3.3 The Power equation

The power equation for the system can be written as;

T2 mj " T6(J06 = T ( U (3.21)

where

T = external torque applied on input l i n k 2

T = the torque transmitted by DO o o Tu) = power necessary to accelerate or to decelerate the system

^ = angular velocity of the shaft to which torque T i s referred

Equation (3.21) corresponds to the dynamical r e l a t i o n :

Rate of work done by external forces = Rate of change of k i n e t i c

energy of the system

or a l t e r n a t i v e l y :

Rate of work done by external forces = Rate of work done by ef f e c t i v e

forces

The variation of the.torques transmitted by r i g i d l i n k 4 and DO^ are shown

i n Fig.(3.3) and i n Fig.(3.4 ) . The corresponding transmitted power i s

shown in F i g (3.5 )

67

E a

x ' C c •H O -< o

l O tj (N •H bO II •H Sj T J

•O a> a> p in :>> £ c

ca <D O (J w •H -H E m (n a c

a 0

a as C7 .M H C O to ' O

b o

CL

o I CD I m

a

ca •H

n /

O O cn

•rl o .to r s i

/

/ \ \

/

O o

o S.

i

C

n

E in c 03

X X o CO

O CD

O •vf

O O O

X "5"

\ i •

o CD

I .

CD

•

C

ro O Q >> n T3 0)

E 0) c

o

cr

o o lO CM

O a in

o

d o

re

05

•a o •H a

•H

hi. •H

to CXI O

i

70

3.4 Graphical Approach

At any i n s t a n t the motion o f any i n d u v i d u a l l i n k i s equivalent t o the

r o t a t i o n of the l i n k as a Vhole about a f i x e d p o i n t i n space. By determining

instantaneous centres of r o t a t i o n , (as shown i n .Figure (3.6 ) g r a p h i c a l

f o r c e a n a l y s i s ) , , i t i s possible t o determine the dynamic forces

a c t i n g on the mechanism. A d e t a i l e d a n a l y s i s o f t h i s procedure i s presented

i n ( 2 1 ) , ( 1 4 ) , ( 1 2 ) , (13) and ( 3 ) .

R e f e r r i n g t o F i g . ( 3 . " f f ) , 0 ab 0 and 0 cd 0 can be t r e a t e d as two

separate but dependent four-bar chains, I and I are the l o c a t i o n s of the

instantaneous centres of r o t a t i o n . Both the l i n e a r v e l o c i t y of the mass

centres and the angular v e l o c i t y of r o t a t i o n vary from i n s t a n t t o i n s t a n t .

The l i n e of a c t i o n o f the fo r c e a p p l i e d t o the l i n k s does not pass through

t h e i r mass centres. The i n d i v i d u a l l i n k s are constrained t o move i n a

d e f i n i t e way by the adjacent l i n k s t o which they are connected, and the

r e s u l t a n t of a l l the forces applied through those connections i s equal t o

the f o r c e r e q u i r e d t o accelerate the l i n k , the e f f e c t i v e f o r c e R . The

magnitudes o f the e f f e c t i v e f o r c e i s replaced by another f o r c e equal t o

-m. a .., displaced from the mass centre a distance h.. This f i c t i t i o u s i c g i x fo r c e replaces the combined e f f e c t s o f the i n e r t i a torque and the i n e r t i a

f o r c e , ab i s a l i n k w i t h pins a t A and B, constrained t o move along the

paths shown. Since the weight of the l i n k s i s small i n comparison t o the

other forces which act on the l i n k , the e f f e c t of g r a v i t y i s

ignored as p r e v i o u s l y . The magnitude and l i n e s o f a c t i o n of R i s

determined. A s i m i l a r procedure i s a p p l i e d t o l i n k cd. The f o r c e F^ which

i s a p p l i e d t o the l i n k AB a t p i n A, by the crank arm O A w i l l have a

component F \ t a n g e n t i a l t o the path of A and also a component F i n perpendicular t o the path o f A. F does the u s e f u l work on the l i n k . F a a i

constrains the pin.. A t o f o l l o w the given path. This assumption i s

s i m i l a r l y v a l i d f o r the forces a c t i n g at b,c and d:. By c a l c u l a t i n g the

72

component i n the t a n g e n t i a l d i r e c t i o n , the component i n the normal d i r e c t i o n

and. the two other components a c t i n g on the next p i n are found from the

e q u i l i b r i u m c o n d i t i o n s of the l i n k , namely, the vector sum of a l l the forces

which act on the l i n k being zero, and the a l g e b r a i c sum of the moments of

the forces about any p o i n t din t h e i r plane being zero. Components of F , a

F , F and F , normal t o the paths of a, b, c and d are known. For l i n k b e d ab, the moments are taken about I , the p o i n t of i n t e r s e c t i o n of the l i n e s

of a c t i o n of F and F and f o r l i n k cd, about I , the i n t e r s e c t i o n b a 1 of the l i n e s of a c t i o n of F and F The equations f o r F * and F *

c d b d aire then

F 1 = "CM } R 3 ( I 2 X ) (3.22)

F d

<M 2>

i R 5( z I ) "

( d i x ) (3.23)

where

R 3 = m 3 3g3 h 3 = ha3 " R3

R 4 = - m4 3g4 h 4 =

" R4

R 5 = " m5 ag5 h 5 = V 5 " R5

R e = - r a6 a

g 6 h e = V e

• " R6

The moment o f i n e r t i a values f o r the s h a f t s are included i n the above

expressions as a p p r o p r i a t e l y . A f t e r the magnitudes of » F^^^, a n d F** are obtained by drawing the f o r c e polygon. The torque which must be d a p p l i e d by the crank arm O^a, to the whole system, i n order t o overcome

the combined e f f e c t s o f i n e r t i a i s given by the product F 0 a. a l

73

As i n the e a r l i e r s e ctions, the crank i s assumed t o r o t a t e a t constant

angular v e l o c i t y

3.5 S i m p l i f i e d Force A n a l y s i s

As mentioned i n s e c t i o n 3.1 n e g l e c t i n g the i n e r t i a e f f e c t s o f l i n k s

3,4,5 and 6, which are r e l a t i v e l y small i n comparison w i t h the .

i n e r t i a e f f e c t s of s h a f t s through 0 and 0 , the f o r c e a n a l y s i s can be

s i m p l i f i e d t o a l a r g e e x t e n t . From Fig.(3.7 ),the f o r c e a p p l i e d t o l i n k

5 by l i n k 6, F , i s determined by t a k i n g moment about 0 o 3

T 6 = - W * 6 = F 5 k l ( 3 ' 2 4 )

where k^ = Moment arm

from equation (3.24)

F 5 =

k l

Repeating the same procedure f o r 0

_ ID2 «4 + F 5 k 2 + F 3 k 3 = ° ( 3 ' 2 5 )

F 5 1 = " F5

F 3 X = " F3

from equation (3.25)

F 3 = I02°'4 " F 5 k 2 k 3

Torque r e q u i r e d t o d r i v e the mechanism i s then,

T 2 = F 3 k 4

Although the a n a l y s i s presented above does not give very accurate

r e s u l t s , i t o f f e r s a quick and reasonable method t o determine the a x i a l

l o a d i n g range on l i n k s 3 and 5.

74

CO

P CO

CO

Li) in L l . I D IX) LL-

CM I CO u :

CM

CO L L , CO

CO bD

CO

I I CO

CO

o

75

3.6 Stress An a l y s i s

The a x i a l s t r e s s v a r i a t i o n f o r l i n k s 3 and 5 are determined from;

a = ^AX a n d a _ F3AX 5AX A 3AX A_

From F i g . (3.9) and (3.10) -5 2

A^ = cross s e c t i o n a l area . - l i n k 3 - BB = 10 x 14.95 m _5 o

A 2 = cross s e c t i o n a l area - l i n k 5 - CC = 10 x 6.32 m

The l i n k s do not have a uniform c r o s s - s e c t i o n a l area thus a x i a l s t r e s s

v a r i e s accordingly. Stresses are p r o p o r t i o n a l w i t h the square of the crank

speed. V a r i a t i o n of a x i a l stresses f o r A and A are shown i n F i g . (3.7;) J. £»

f o r a complete c y c l e , a t a crank speed o f 2500 r.p.m.

The peak t o peak (maximum t o minimum) s t r e s s values are r e f e r r e d t o

as st r e s s range, and the v a r i a t i o n of t h i s range w i t h respect t o the square

of the crank-arm angular v e l o c i t y i s given i n F i g . ( 3 . 8 ) . Plane bending

s t r e s s values f o r the same cross-sections o f l i n k s 3 and 5 can be c a l c u l a t e d

from M_y . F _ . \ l /2) y _ ? 5 Jc5 _ 5b \ 5 Jc5 , . /

a = ~1 " 7 l (3.26) 5b c5 ^ " c5

where y , = distance from the n e u t r a l a x i a l - c r o s s - s e c t i o n CC Jc5 I _ = Moment of i n e r t i a of the cross-section CC c5

and

0- = - S - b l = 3b 3 b3 (3.27) 3 b XB3 ^ 3

where y^„ = distance from the n e u t r a l a x i s - c r o s s - s e c t i o n BB bo I„„ = Moment of i n e r t i a o f the cross-section BB

and F = f o r c e component a c t i n g normal t o l i n k 5 5b F = f o r c e component a c t i n g normal t o l i n k 3 3b

76

110 N/m

100

i * i \ / \ /

/ • \ \ 7 \ / \ / \ / 20

7 o I

A 200 100 300

I V

ink o

l i n k 3 60

/

/

/ /

110

F i g . 3.7 V a r i a t i o n o f a x i a l stresses i n l i n k s 3 and 5 w i t h the crank angle (crank speed = 2500 r.p.m.)

77

s t r e s s _ r a n g e _ N / m « 1 0

i

500

4 0 f l

3 0 0

200

100

10 ( r a d / s ) 2 ( a n g u l a r v e l o c i t y ) *10

f

F i g . 3.9 T h e o r e t i c a l v a r i a t i o n o f a x i a l s t r e s s range by the square of crank-speed

78

The l i n k s are t r e a t e d t o be beams of non-uniform c r o s s - s e c t i o n , and

magnitude of s t r e s s and d e f l e c t i o n i s assumed t o be d i r e c t l y p r o p o r t i o n a l

t o the load.

Resultant s t r e s s over a cross-section i s then given by:

nax My cn

- I cn (3.28)

79

LINK 3-connecting rod

,-D=U.985

T 22.5 L

A +B *

7 n B.B cross.sAA

Oi e >0 r

D = 22 69.2

i

A l l dimensions are i n mm

F i g . 3.9 Link 3

!

so;

LINK-5

f ron tv iew

i

cross s e c c c

scale. x2 !

• F i g . 3.10

81

0

0 01

0)

0> i n

a ii

ho

CD C3 10

a> 0)

in •H

CO

to CNJ

V. \ o

to

CM

<r /vi i O ^ <r to 0 4 i i ^ i i

82

11 NK- 5

P 'J| r e a X i! a I f o r c e

Figure 3.1JS V a r i a t i o n o i axin.l crank angle 2500

83

P u r e a x i a l f o r c e

200° .1 00

/ \ ./

2500rp

LINK-3

Figure 3.13 V a r i a t i o n of a x i a l f o r c e a t crank angle

I

84 1 aJ- W:

500 F i g . 3.14

i.

Theoretxcal maximum a x i a l t e r t s i l 500

s t r e s s

10C

.'•DO 3000 -.000 to peak s t r e s

100 r.p.mr •

200

Theoretical maximum a x i a l

compressive s t r e s s 300

Link 5

Link 3

6 0 0 -

85

Chapter 4

Instrumentation, Experiments and Experimental Results

4.1 Instrumentation

The s t r a i n gauge i n s t a l l a t i o n and a n c i l l a r y equipment i s shown i n

Figs. ( 4 . 1 ) , ( 4 . 2 ) , P l ate (4.1) and P l a t e ( 4 . 2 ) . Each p a i r of gauges i s

w i r e d i n various c o n f i g u r a t i o n s on the extension box t o measure

only a x i a l . and only . bending (z d i r e c t i o n ) s t r a i n s . The

output from the extension box i s fed i n t o the c a r r i e r - a m p l i f i e r where

a (met re i S used i n con j u n c t i o n w i t h the balance knobs on the extension ^»»

box t o balance the bridge c i r c u i t . The output from the a m p l i f i e r i s fed

through a lead t o an u l t r a - v i o l e t recorder where dynamic records are

produced on p h o t o - s e n s i t i v e paper. To prevent f a i l u r e due t o overloading

r e s i s t e r s and a fuse are connected i n the leads t o the u.v. recorder. I f

a record was not r e q u i r e d i t was convenient t o d i s p l a y the response o f the

s t r a i n gauges on the o s c i l l o s c o p e t o examine the s t r a i n pattern.A s l o t t e d metal

d i s c i s attached t o the d r i v e s h a f t and an electro-magnetic pick-up u n i t

i s placed close t o the d i s c t o produce a v a r i a b l e voltage. This v a r i a b l e

voltage showed one b l i p f o r each r e v o l u t i o n of the s h a f t which i s recorded

by the u.v. recorder a t the same time as the t r a c e from the s t r a i n gauges

was recorded, which enabled the speed o f the s h a f t t o be determined, f o r

each recording. A variable-speed 2HP e l e c t r i c motor i s used t o d r i v e the loom

P l a t e ( 4 . 2 ) . The wires l e a d i n g from the t a g s t r i p s on the specimens t o the

extension box were p a r t i a l l y p r o t e c t e d by p l a s t i c tubes against f a t i g u e

f a i l u r e . I n order t o prevent damage due t o f r a c t u r e at h i g h o p e r a t i n g speeds,

a sheet s t e e l cover i s used over the combs. The basic p r i n c i p l e a p p l i e d i n

the design o f the dynamic measuring system i s t h a t the whole set-up w i l l

f a i t h f u l l y measure the s t r a i n no matter how i t v a r i e s w i t h time. Since

the s i g n a l from the strain-guage c i r c u i t s i s small a c a r r i e r a m p l i f i e r i s

engaged, which i s capable of r e c i e v i n g s i g n a l s i n the m i l l i v o l t range and

8&

X t — . : CL LU <

cc o o ' > •

1 _ l < ..._j • cr — •

i CO => o

o u

CC LU

LU U_

_ l cr Q_ < o <

i \

z

o

co X

LU o . t — X LU

CD

cn i

._ cn

if)

0> U)

1 CL cn r) a i £ u 1 o CL

< O — c

a> CD C

e a u M « •a

o o i-i m

3 I P

t-> c <D E •H o a •a w

bO •H

87

I

CO X LU

CO a O LU CO Q •7 CO UJ

co o CL CL

I 7;

o CO UJ LU

CO DC CO UJ 1 1

I UJ

GC fC LU

t—

LU

i l l

UJ

CO LU

UJ o cc or. u. cc O

to

89

' 1

r

t i * Li I

P l a t e 4.1 O s c i l l o s c o p e , a m p l i f i e r , s w i t c h box, u.v. r e c o r d e r ( w i t h speed p i c k - u p c o n n e c t i o n )

s t r a i n gauge le a d s s t r a i n gauge l e a d s

89

ft

1 s i

i l l

r •r

I

P l a t e 4.3 2 HP E l e c t r i c motor and speed v a r i a t o r u n i t

90

of s u p p l y i n g a s i g n a l i n the v o l t range. The r e c o r d i s on s e n s i t i z e d paper

which r e q u i r e s no developing. A p o l a r o i d land-type camera i s f i t t e d to

the o s c i l l o s c o p e to photograph the experiemntal s t r a i n p a t t e r n . The r e s u l t s

a r e a l l taken from records produced by the u.v. r e c o r d e r . I t i s assumed

t h a t the only r e s i s t i v e elements i n the s t r a i n - g a u g e c i r c u i t a r e the gauges

themselves ( t h e presence of l e a d s which a f f e c t s e n s i t i v i t y and c a l i b r a t i o n

a r e n e g l i g a l a l e ) .

4.2 Elements of the Experimental Set-up.

1. C a r r i e r - A m p l i f i e r ; A U n i v e r s a l c a r r i e r - a m p l i f i e r , type 581 DNH-Peekel,

i s used. E l e v e n gauge f a c t o r s were a v a i l a b l e , ranging from 1.75 to 2.25,

w i t h an accuracy of 0.5%. H a l f and q u a r t e r - b r i d g e (120 Q) c i r c u i t s a r e

employed. There a r e s i x i n t e r n a l c a l i b r a t i o n v a l u e s , 30, 100, 300, 1000,

3000 and 10000 m i c r o s t r a i n . Bridge v o l t a g e i s 5 v o l t s , on the most

s e n s i t i v e range (3 JJ,S , f u l l - s c a l e d e f l e c t i o n ) f o r one a c t i v e s t r a i n - g u a g e .

The output g i v e s 1 v o l t f o r f u l l s c a l e of the metre and remains l i n e a r

f o r dynamic measurements even up to 10 v o l t s . The s p e c i f i e d a c c u r a c y of

the equipment f o r dynamic measurements i s + 0.75% f o r a l l ranges. Time

delay i s 0.5 ms f o r 5000 Hz. L i n e a r i t y of the r e c o r d e r output i s + 0.05%

f o r 2 v o l t s peak-to-peak output and + 0 . 1 % f o r 20 v o l t s peak-to-peak

output, provided t h a t the input i s balanced f o r c a p a c i t y . D r i f t i s approx­

imately 1 p, s t r a i n / d a y

2. U l t r a - V i o l e t Recorder; A "Southern Instruments" u l t r a v i o l e t o s c i l l o g r a p h

10-100 s e r i e s i s used f o r continuous d i r e c t r e c o r d i n g of input s i g n a l s .

S i g n a l i s fed to a minature t u b u l a r galvanometer which r e f l e c t s a spot-.of

i n t e n s e u l t r a v i o l e t l i g h t onto the p h o t o s e n s i t i v e r e c o r d paper. The def­

l e c t i o n of the galvanometer i s a f u n c t i o n of the amplitude of the i n p u t -

s i g n a l c u r r e n t . A c h o i c e of e i g h t paper speeds i s a v a i l a b l e . Two of the

input s o c k e t s a r e used. F i r s t one f o r the c a r r i e r - a m p l i f i e r o u t p u t - s i g n a l

and the second one f o r the e l e c t r o m a g n e t i c pick-up u n i t o u t p u t - s i g n a l . A

91

paper speed of 37 mm/s i s used f o r c a l i b r a t i o n w h i l e paper speeds of 111

and 333 m/s a r e a l t e r n a t i v e l y used dueing a c t u a l measurements. Four f i x e d

time i n t e r v a l s a r e a v a i l a b l e , 0.01, 0.1, 1 and 10 seconds, and the s p e c i f i e d

a c c u r a c y of t h e p r i n t e d l i n e s i s w i t h i n +• 2%. Paper speed s t a b i l i t y i s + 3%.

S i n c e , i n s t e a d o f pen-aim, an u l t r a v i o l e t l i g h t beam of zero mass i s i n t r o ­

duced, d i r e c t r e c o r d i n g o f f r e q u e n c i e s up to 10 KHz can be performed. SMI,

Type SMI/N, S e r . No. 1480-2 galvanometer i s employed f o r r e c o r d i n g .

3. E x t e n s i o n Box: A "Peekel 4-channel 4 UD" type s w i t c h box i s used.

There a r e four input channels, f o r f u l l - b r i d g e , h a I f - b r i d g e or q u a r t e r -

bridge, 120 o c o n f i g u r a t i o n s . Each channel has four b a l a n c i n g c o n t r o l s ,

t h r e e f o r r e s i s t i v e b a l a n c i n g and one f o r c a p a c i t i v e b a l a n c i n g of the

bridge c i r c u i t . The s p e c i f i e d t y p i c a l e r r o r of s w i t c h c o n t a c t s i s 1 - 2

m i c r o s t r a i n . Ranges of r e s i s t i v e and c a p a c i t i v e b a l a n c i n g a r e + 6000 p,s

and + 1000 pf r e s p e c t i v e l y .

4. O s c i l l o s c o p e : A cathode ray o s c i l l o s c o p e i s employed to d i s p l a y the

r a p i d l y v a r y i n g s t r a i n waveform. S i n c e the output of the s t r a i n gauge

c i r c u i t i s connected to v e r t i c a l a m p l i f i e r s , v e r t i c a l d e f l e c t i o n i s r e l a t e d

to s t r a i n . The i n t e r n a l sweep c i r c u i t a l l o w s the t r a c e to be d r i v e n h o r i ­

z o n t a l l y a t a p r e s e l e c t e d r a t e .

4.3 S t r a i n gauges and gauge i n s t a l l a t i o n s 5 i

S i n c e s t r a i n gauges a r e used to measure v a r y i n g and repeated s t r a i n s ,

s p e c i a l c a r e i s e x e r c i s e d i n gauge s e l e c t i o n , i n gauge bonding and i n l e a d . *

attachment. "Timsley, T e l c o n " elements type 7/120/EC, of d i f f e r e n t batch i

numbers, gauge f a c t o r = 2.18 Range = 120 q + 1%, electrical..' s t r a i n gauges a r e

employed. The r e s i s t i v e element i s f i x e d to a tr a n s p a r e n t p l a s t i c base,

and i s bonded to the po i n t a t which s t r a i n i s measured. The gauges a r e

pla c e d on the specimens as shown i n F i g . ( 4 . 2 ) . Although i t was f e l t t h a t

h e l i c a l l y wound w i r e gauges and " I s o e l a s t i c " w i r e gauges o f f e r a s u p e r i o r

92

f a t i g u e l i f e , r e c e n t r e s e a r c h has shown th a t other standard gauges have as

great or even g r e a t e r f a t i g u e l i f e ( 6 ) . F l a t - g r i d , w i r e and f o i l

metal gauges on paper back and bonded w i t h c e l l u l o s e n i t r a t e cement have

been s t r a i n c y c l e d (+ 1000 p,s / i n f o r 1,250,000 c y c l e s without f a i l u r e

( 6 ) ) . Epoxy-backed f o i l metal gauges bonded w i t h epoxy cements a r e

capable of more than 300,000 c y c l e s a t + 1500 ^ s . However i t i s g e n e r a l l y

agreed t h a t as the magnitude of c y c l i c s t r a i n i n c r e a s e s from 1500 J J , S ,

the f a t i g u e l i f e i s reduced very r a p i d l y ( 6 ) . A w e l l e s t a b l i s h e d

f a c t i s t h a t the connection of the l e a d w i r e s i s one of the most c r i t i c a l

s t e p s i n the i n s t a l l a t i o n of a gauge f o r c y c l i c s t r a i n s e r v i c e .

C y a n o a c r y l a t e ( L o c k t i t e , I S - 12 a d h e s i v e ) i s used to bond the base to

the specimen. However s i n c e i t i s a poor gap f i l l e r the s u r f a c e of

the specimen i s f i l e d to o b t a i n a smooth area and i s cleaned w i t h carbon

t e t r a c h o l o r i d e and/or acetone. Dove and Adams ( 6 ) have used cyano-

a c r y l a t e to measure dynamic s t r a i n s ( r i s e time 1 m i l l i s e c o n d ) as high

as 2000 |j,s. C y a n o a c r y l a t e cement bonds under the a c t i o n of modest

contact p r e s s u r e , and works s a t i s f a c t o r i l y up to temperatures of 120°C.

Cement cur e s by chemical p o l y m e r i z a t i o n , and o b t a i n s i t s optimum

adhesive s t r e n g t h a f t e r about 48 hours. I n v i b r a t i o n environments

the weak po i n t of the gauge i s the point a t which the f i l a m e n t i s

connected to the gauge l e a d . The gauges a r e checked f o r c o r r e c t bonding,

gauge r e s i s t a n c e and guage-to-specimen r e s i s t a n c e by u s i n g a low-voltage

type of mega ohm metre. The t r a n s v e r s e s e n s i t i v i t y of the gauge i s

n e g l i g i b l e , and i t i s assumed t h a t , s i n c e the specimens have a r e l a t i v e l y

high modulus of e l a s t i c i t y , the s t r a i n i s uniform, through the c r o s s -

s e c t i o n s a t which the gauges a r e a p p l i e d . The minor e f f e c t of l o c a l i s e d

s t i f f e n i n g on the s u r f a c e , producing d i s t o r t i o n of the s t r a i n p a t t e r n

i s n e g l e c t e d . The gauge f a c t o r i s taken to be c onstant,

s i n c e p r a c t i c a l l y i t i s unchanged even i n the p l a s t i c region. U n l e s s , thermal

or mechanical shock i s expected, c y a n o a c r y l a t e cement i s regarded as s u p e r i o r

93

t o other gauge bonding m a t e r i a l s f o r dynamic s t r a i n measurements. The o i l

temperature does not vary i n s h o r t term operations and i s assumed to s t a y

c o n s t a n t .

The bonded gauges were f i r s t p r o t e c t e d w i t h A r a l d i t e a g a i n s t o i l . How­

ever o r d i n a r y A r a l d i t e c o a t i n g i s a t t a c k e d by the o i l (Mobil Vactra o i l ) and

f a i l e d to hold the i n s t a l l a t i o n . T h i s major i n s t a l l a t i o n problem i s s o l v e d

by u s i n g "Micro Measurement I n c . M-Coat G Compound". I n order to be a b l e

t o measure the s t r a i n s a c c u r a t e l y , gauge i n s t a b i l i t y and mechanical damage

a r e minimized. B a s i c mechanical damages were due to v i b r a t i o n , high

o p e r a t i n g speed and l i m i t e d a v a i l a b l e space. During measurements s t r a i n -

gauge i n s t a l l a t i o n s have been r e p l a c e d s e v e r a l times because of mechanical

damage and i n s t a b i l i t y . O i l a c t t o reduce the gauge-to-surface r e s i s t a n c e

or p a r t i a l l y to s h o r t - c i r c u i t s e c t i o n s of the gauge i t s e l f . The e f f e c t of

o i l i s to p l a c e a r e s i s t a n c e path i n p a r a l l e l w i t h the s t r a i n gauge, prod­

u c i n g a change i n r e s i s t a n c e e q u i v a l e n t to s t r a i n . T h i s e f f e c t i s very

important s i n c e the change a s s o c i a t e d w i t h s t r a i n i s very s m a l l . The

c o a t i n g m a t e r i a l employed i s a two-part 100% s o l i d s p o l y s u l p h i d e modified

epoxy compound, which provides a tough f l e x i b l e l a y e r , o f f e r i n g good

p r o t e c t i o n a g a i n s t commercial o i l s , greases, g a s o l i n e , most a c i d s , a l k a l i s

and most s o l v e n t s . I t cures to a f i r m , tack f r e e c o n d i t i o n i n s i x hours

a t 24°C. F u l l cure i n 24 hours again a t 24°C. I t can be s a f e l y used up to

an o i l temperature of 82°C. An incomplete p r o t e c t i o n around l e a d w i r e s was very

f r e q u e n t and a common cause of o i l p e n e t r a t i o n i n t o the gauge i n s t a l l a t i o n s . By

i n t r o d u c i n g p r a c t i c a l s o l u t i o n s , and g e t t i n g f a m i l i a r w i t h the v i t a l

problems c r e a t e d by h o s t i l e environment, the number of i n s t a l l a t i o n

f a i l u r e s a re reduced, and longer terms of s t a b i l i t y and a c c u r a c y i n readings

a r e achieved. Coatings a r e a p p l i e d t o cleaned s u r f a c e s , s i n c e c o a t i n g s

extending i n t o unclean a r e a s w i l l loosen w i t h time. G e n e r a l l y a t h i c k

c o a t i n g o f f e r s a more d i f f i c u l t path f o r o i l p e n e t r a t i o n than a t h i n one.

M-Coat G forms a f l e x i b l e rubbery c o a t i n g , and i t s chemical r e s i s t a n c e may

94

be f u r t h e r improved by a one hour bake a t 93° C. The s o l d e r j o i n t s and

w i r i n g t e r m i n a l s a r e covered w i t h a t h i n l a y e r of M-Coat D (an a i r - d r y i n g

s o l v e n t thinned a c r y l i c c o a t i n g ) before a p p l y i n g M-Coat G i n order to

prevent any e l e c t r i c a l leakage under adverse c o n d i t i o n s . The prime coat

d r i e s i n 15 minutes under normal ambient c o n d i t i o n s . I n a p p l y i n g M-Coat G

c a r e i s taken to avoid a i r pockets and the c o a t i n g i s extended out as f a r

as p o s s i b l e on a l l s i d e s beyond the edges of the M-Coat D l a y e r . O i l , most

of t e n , e n t e r s coated s t r a i n - g a u g e i n s t a l l a t i o n s along i n s u l a t e d l e a d w i r e s

producing s i g n a l s which a r e u n r e l a t e d to, but h a r d l y d i s t i n g u i s h a b l e from s t r a i n .

F or t h i s reason w i r e s a r e completely coated as f a r back from the i n s t a l l a t i o n

as p r a c t i c a l , which i s u s u a l l y not more than one cm, due t o space l i m i t a t i o n s .

However, most of the time i n s u l a t e d w i r e does not bond w e l l to the c o a t i n g .

4.4. Experiments

1. Dynamic S t r a i n Measurements

As shown i n P l a t e ( 4-3 ) , a v a r i a b l e speed (480-4320 output, r.p.m.),

2 HP, 3 phase 50 c y c l e s e l e c t r i c motor i s used to run the loom, r e p l a c e d

by the o r i g i n a l 1.1 Kw, 1410 r.p.m., 1.5 HP, 3 phase 50 c y c l e s motor. A

tachometer i s used t o check d r i v e s h a f t speed i n a d d i t i o n to the u.v.

r e c o r d e r . The motor i s c a l i b r a t e d a g a i n s t the d r i v e s h a f t speed by u s i n g

the v a r i a c speed s e l e c t o r unit- The bending s t r a i n s measured a r e

i n the z d i r e c t i o n , p e r p e n d i c u l a r to t h e x-y plane of the mechanism. The

s t r a i n s due to plane bending a r e not measured, ( a s e x p l a i n e d i n

the i n t r o d u c t i o n , see S e c t i o n 1 ) .

The r e s u l t s of the dynamic s t r a i n measurements a r e given i n Table ( 4 . 2 ) .

Measurement p e r i o d s , u s u a l l y d i d not exceed 15 minutes. The d r i v e s h a f t

speed i s g r a d u a l l y i n c r e a s e d from approximately 1050 r.p.m to 2550 r.p.m.

The c i r c u i t i s rebalanced f o r r e s i s t a n c e and c a p a c i t a n c e before each run,

and a c a l i b r a t i o n t r a c e has been recorded. The b a s i c problems encountered

during dynamic-measurements a r e :

95

a - Gauge i n s t a l l a t i o n f a i l u r e s

b - E x c e s s n o i s e and v i b r a t i o n due to improper contact of b e v e l

gears at high o p e r a t i n g speeds

c - Mechanical f a i l u r e due to s l i p between l i n k 6 and s h a f t 0

d - s t r e t c h i n g and bending of the l e a d w i r e s

2 S t a t i c B e n d i n g - s t r a i n measurement

To examine the nature of bending s t r a i n s , a s t a t i c bending s t r a i n t e s t

i s performed on the mechanism. F i g . ( 4 . 3 ) shows the simple s e t up. A

0.3 m l e v e r arm i s a t t a c h e d to the d r i v e s h a f t t o apply torques of known

magnitude to the system. The t e s t r e s u l t s a r e given i n t a b l e (4.3 ) . The

s h a f t Og i s locked by a clamp mechanism to a l l o w the f o r c e s to be t r a n s ­

mitted throughout the l i n k s . Both a x i a l and bending s t r a i n s ( z d i r e c t i o n )

a r e measured under the same l o a d i n g c o n d i t i o n s .

The v a r i a t i o n of a x i a l and bending (z d i r e c t i o n ) s t r e s s e s w i t h the app­

l i e d torque i n each l i n k a r e shown i n f f i g s . (4.4) and (4.5). A p p l i e d torque

i s i n c r e a s e d up to 74 N.m. Half - b r i d g e (120 0) c i r c u i t s a r e employed during

measurements. The crank angle ' p s i ' i s kept 90° (from the h o r i z o n t a l i n

c o u n t e r - c l o c k w i s e d i r e c t i o n ) .

3 S t r e s s d i s t r i b u t i o n i n l i n k s 3 and 5

Using the same experimental set-up, s t r a i n v a r i a t i o n w i t h i n the l i n k s

themselves a r e measured. Strain-gauges on opposing f a c e s of l i n k 5 a r e i d e n t i f i e d

as gauges m and k. Quarter-bridge (120 p) c i r c u i t i s employed to measure

the combined a x i a l and bending s t r a i n on each f a c e s e p a r a t e l y by feeding the

two l e a d - w i r e s d i r e c t l y i n t o the c a r r i e r a m p l i f i e r . The procedure i s repeated

f o r both gauges by r e v e r s i n g the l i n k . During the measurements crank angle

' p s i ' i s kept a t 90°. A maximum torque of 35.3 N.m i s a p p l i e d to the main

d r i v e s h a f t v i a the lever-arm. The pure a x i a l compressive s t r a i n i s then

96

65

CD

CO

bO

CQ co CD

CO i

CD

CO CD

CO CO Co CD

CO

CO

D)

CD

CO

CO

CO

CNI

CD

CD

CD

S t r e s s . 1 0 ° N/ m

2 0 0 0

50

T o r q u e - N m

F i g . 4.4 S t a t i c Bending - s t r a i n t e s t f o r l i n k 5. V a r i a t i o n of a x i a l and bending ( z - d i r e c t i o n ) s t r e s s e s w i t h the a p p l i e d torque

S t r e s s . 1 0 5 N / m 2 98

f

5oq

4 0 0

i

20

100

A x i a l 5 0 B e n d i n g

70 5 0 3 0 10 T o r q u e - Nl-m

• F i g . 4.5 S t a t i c bending s t r a i n t e s t f o r l i n k 3 - V a r i a t i o n of a x i a l and bending ( z - d i r e c t i o n ) s t r e s s e s w i t h the a p p l i e d torque.

99

measured by u s i n g a h a l f - b r i d g e c i r c u i t . S t r a i n gauges on opposing f a c e s of

l i n k 3 a r e i d e n t i f i e d as gauges c and d, and the above procedure i s

s i m i l a r l y a p p l i e d to determine the s t r a i n v a r i a t i o n . R e s u l t s a r e shown

i n F i g u r e s (4.6) and ( 4 . 7 ) .

4. T e n s i l e T e s t

Separate t e n s i l e t e s t s a r e performed on l i n k s 3 and 5. A "Servomex

C o n t r o l s L t d . , tensometer, m e t r i c type-E" i s used i n combination w i t h a

"Peekel 4-channel 4 UD" e x t e n s i o n - box and a "Peekel type 581 DNH" U n i v e r s a l

c a r r i e r - a m p l i f i e r . The t e s t machine i s provided w i t h a u n i t which automat­

i c a l l y draws a t e n s i l e t e s t diagram r e p r e s e n t i n g the r e l a t i o n between the

load and the e x t e n s i o n . Two s t r a i n gagues a r e f i x e d on opposing f a c e s of

the l i n k s , connected to the extension box. A h a l f - b r i d g e 120 Q c i r c u i t i s

used to measure the t e n s i l e s t r a i n . A 2500 N c a p a c i t y l o a d c e l l i s employed.

Motor speed was 600 r.p.m.and the cross-head speed tfas 9 mm/inin. A p a i r of

s p e c i a l l y manufactured chucks a r e used f o r each l i n k to g r i p them between

the moving c r o s s head and the base. The chucks were> f i t t e d w i t h heat t r e a t e d

s t e e l p i n s , having diameters as c l o s e as p r a c t i c a l to the i n n e r end bore

diameters of the l i n k s . The chucks were designed to ensure a c e n t r a l a p p l i c ­

a t i o n of the load. The d i r e c t i o n of the paper movement \ws the_same.as t h a t

of the crosshead. The event marker s w i t c h i s used from time t o time to

record the a c t u a l a p p l i e d load a t c e r t a i n s t r a i n v a l u e s r e g i s t e r e d on the

c a r r i e r a m p l i f i e r . The moduli of e l a s t i c i t y and y i e l d p o i n t s t r e n g t h s a r e

determined from the experimental data shown i n F i g . 4.8 f o r c r o s s - s e c t i o n s

BB and AA of the l i n k s 3 and 5 r e s p e c t i v e l y .

100

s t r U f

ga iJc je - m •A

a x i a l ' v b f e h

i LI N.K-5 7 0 0

5 0 0 i

i CL

200 a.

30 10 n To rq ue. N_m

30 0

5 0 0 i

i 70 0

gauge-m 1 0 0 0 1

s4 i I I

12 00

c a s e.T c a s e»2 (I j n k. r e v e r s e td ) •

F i g . .4.6;

101

us_ axial +be'nding

i

500 gauge_d

L I N K-3

400

gauge_c

300

00

100

Torque Nm 0 I S 25 35

100

200

300

F i g . 4.7 i

24:K0Vi

102 CO

03 CO

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V) 0

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in CD

CO (/) 00 CO TO CD CU 0)

CO l bo CN

II I I L O CO LU O CO CO 0)

CO CD

CM

103

4.5 Dynamic S t r a i n Measurement Results

Since two a c t i v e gauges are connected i n a h a l f - b r i d g e c i r c u i t f o r

each set of r e s u l t s , the reading from the bridge i s twice t h a t which would

be expected i f only one a c t i v e gauge were used. The r e s u l t s are t h e r e f o r e

d i v i d e d by two t o ob t a i n the r e a l s t r a i n and s t r e s s values shown i n Table (4.2)

The v a r i a t i o n of the peak t o peak stresses (both a x i a l and bending) w i t h

the crank speed f o r l i n k s 3 and 5 are shown i n Figures 4..9 t o 4. 13

Experimental sample s t r a i n p a t t e r n s obtained, are shown i n F i g . 4.14, a - f f o r

various crank speeds. Experimental v a r i a t i o n of the c y c l i c a x i a l stresses

i n both l i n k s w i t h the crank angle .. are presented i n Figures 4.1'5and 4.16.

4.6 Speed F l u c t u a t i o n

Crank arm speed i s assumed t o remain constant d u r i n g a measurement.

However the f o l l o w i n g s i m p l i f i e d t h e o r e t i c a l c a l c u l a t i o n based on t o t a l

k i n e t i c energy of the system i s made t o p r e d i c t an e s t i m a t i o n ;

Let

= Moment of i n e r t i a o f the main d r i v e s h a f t w i t h the flywhe e l

I = Moment o f i n e r t i a o f s h a f t 0 o

I g = Moment of i n e r t i a o f sha f t 0^

From F i g . 3.6 (Torque diagram, f o r the crank speed = 2500 r.p.m.), Let o

be the angular v e l o c i t y of the crank arm at the crank angle . ijf = 50 . Also l e t Q. and 0- be the angular v e l o c i t i e s of s h a f t s 0 and 0„ at the

2 2 2

same crank angle. Then 2 ( T o t a l K i n e t i c energy) + *2^4 + I3^6 = c o n s t a n t

or

2 (K.E) 2 -„ . " • i - — — » - ( 4 : i ) 02

104

The v a r i a t i o n o f are shown i n Table 4.1 f o r various crank angles, 0°-^360°

by 10° i n t e r v a l s a t an average crank speed o f 2500 r.p.m. = 261.798 rad/s.

From t a b l e 4.1, c o e f f i c i e n t of speed f l u c t u a t i o n i s ;

^nax " °min 2 6 7 - 4 3 ~ 2 5 3 - 9 5 C = = 0.0515

S 0 261.798 av

speed f l u c t u a t i o n i s also measured experimentally by using the u.v recorder

and the speed pick-up u n i t i n combination as explained i n 4.1. The

experimental average value o f c o e f f i c i e n t o f speed f l u c t u a t i o n obtained

was 0.082

1.05

Table 4.1

crank angle p s i (degrees) °2 rad/s

crank angle p s i (degrees)

o 2

rad/s

0 263.95 190 267.43

10 266.68 200 267.41

20 267.43 210 267.19

30 266.26 220 266.88

40 263.82 230 266.56

50 260.93 240 266.25

60 258.29 250 265.91

70 256.39 260 265.39

80 255.58 270 264.53

90 255.99 280 263.10

100 257.50 290 260.96

110 259.64 300 258.26

120 261.81 310 255.61

130 263.52 320 253.95

140 264.63 330 254.12

150 265.35 340 256.35

160 265.97 350 260.05

170 266.60 360 263.95

180 267.14

5 2 s t r e s s . 10 N / m

24 0

2 0 0

150

100

50

106

1. Theoretical

2. Experimental

-r-S> 1000 1 500 2000

crank "speed r;'p:m; 2 5 0 0

F i g . 4.9 T h e o r e t i c a l and experimental v a r i a t i o n of a x i a l s t r e s s range (peak t o . peak s t r e s s ) i n l i n k 5 w i t h the crank -speed.

1 0 " N/m 107

3oa

25a

200

I

150

no

•000 1500 " ' ' 2000' ' ' ' 2500 crank speed r.p.m

F i g . 4.1Q Experimental v a r i a t i o n of bending (;&-direction) peak t o peak stress ( s t r e s s range) i n l i n k 3 w i t h the crank speed.

108 s t r e s s . 1 0 5 N / rn 2 I

F i g . 4.11 Experimental v a r i a t i o n of bending ( z - d i r e c t i o n ) peak t o peak st r e s s ( s t r e s s range) i n l i n k 5 w i t h the crank speed

i

700

500

300

M'OO

I 1000 ,V5;Q0 2 0.00 25 00 crank speed r P m

1 0 5 N / m 2 109

T H E O R E T I C A L

-o_ E X P E R I M E N T A L

1000 1500 2000 " r~250Q CRANK SPEED. R P M

F i g . 4i12. T h e o r e t i c a l and experimental v a r i a t i o n of a x i a l peak t o peak st r e s s (Stress range) i n l i n k 3 w i t h the crank speed

iCTN/m 110

T h e o r e t i c a l , a x i a l l i n k 3 a Experimental a x i a l l i n k 3 b 600 I h c o r e t i c a l a x i a l l i n k 5 Experimental a x i a l l i n k 5 Experimental bending l i n k 5

f - Experimental bending l i n k 3

500

400

30G

f

/ 20C

100

2 ..+4 (ang-vel) *1 0 ( r a d / s )

7 0

F i g . 4.13, V a r i a t i o n of peak t o pt\nk stresses i n l i n k s .1 dud'5 w i n tho sqiuiro of the crank speed

i n ;

F i g . 4.14 Experimental s t r a i n p atterns

:a Record 80. Link 5. Experimental bending ( z _ d i r e c t i o n ) s t r a i n r j a t t e r n . Crank speed = 1573 r.p.m. & Bridge ( 1 2 0 o ) 2 c i r c u i t . Peak t o peak s t r a i n = IS7.4 us, Peak t o peak st r e s s = 205 x 10 5 N/m

b Reco'rd 99. Link 5. Experimental bending ( z - d i r e c t i o n ) s t r a i n p a t t e r n . Crank speed = 2023 r.p.m. £ Bridge (120O) c i r c u i t . Peak t o peak s t r a i n = 417.5 u« Peak t o peak st r e s s = 433.8 x 10& N/m

112

AUS

wmvm 5 H r- l-cycle

c Record 131 Link 5 Experimental a x i a l s t r a i n p a t t e r n . Crank speed = 2130 ' \ Bridge (.1200) c i r c u i t . Peak t o peak s t r a i n a 203.9 |j,s. Peak t o peak

str e s s = 211.8 x 105. N/m

AUS

d Record 16. Link 5. Experimental a x i a l s t r a i n p a t t e r n . Crank speed = 1634 r.p £ Bridge (120n) c i r c u i t . Peak t o peak s t r a i n = 86.7 p,s, Peak t o peak s t r e s s = 90 x 10 5 N/m

I : " 113

>,US

j « m

9' Uncord '19 Link 3. Exporime-tital bending (z d i r e c t i o n ) s t r a i n pat t o r n . Crank speed - 1770 r . p . j n . f t Bridge (126 ) c i r c u i t . Peak t o peak s t r e s s 291.2 x 10 J N/m

US

Record 13. Link 3. Experimental a x i a l s t r a i n p a t t e r n . Crank spoed - 1424 r.p | Bridge (120 ) c i r c u i t j, Pegk t o peak s t r a i n = 48.5 us, peak t o peak s t r e s s = *38.8 x 10° N/m

f 1 0 5 N / m 2

114

Ad

30

20

10

K

0'

T H E O R E T I C A L

E X P E R I M E N T A L

90

• A

. A

270 .360

crank speed=142A r.p.m

F i g . 4.15 Experimental and t h e o r e t i c a l v a r i a t i o n o f ' a x i a l s t r e s s i n l i n k 3 w i t h the crank angle p s i a t a crank speed of 14242r.p.m.- (cross

' s e c t i o n a l area subject t o st r e s s = 14.95 x 10~5 m )

JktO N / m A X I A L S T R E S S

II' \ \ I W I /

t

. • l i

'jf . T H E O R E T I C A L ^ I

11 '. EXPERIMENTAL /

J c r a n k speed =1534 r.p.m | j

u M 0°

ESI 90° 270° 180° 360° » s ,»:-J^*i- 4^*§*..Theoretical and experimental v a r i a t i o n q f . a x l a l j s t r e s e - l n ^ i n k - 5 ^ J ^

•• . — - v i t h the crank angle psi,- at a .crank speed of* 1534r. pTinlU.-.' ~

116

CM

...m/N o 0 l * 6 S

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1089

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1473

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r m CO

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s 00 rH rH

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•ui'd^a .xaiauioqoBj,

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1560

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1980

1320

1680

1080

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1800

1980

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d n - j j o x d paads .iapjooaj " A n 14

16

1517

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1709

1780

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1256

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1770

1942

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in CO

t £ T-t T-t

co m CO

m CO

CD m CO

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=

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2040

2160

=

1560

1800

1920

2040

2160

2040

1080

1320

1440

•ui'd'.i • T B U 3 T S dn-jpxd paads

j o p j o o a a *An 2034

2023

1931

1517

1785

1942

2136

2124

1942

1062

1089

1214

1416

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t r

111

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1 T-t CO

i CM CO

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1 CD CO

1 t-CO

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119

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131

Record No. - l

A p p l i e d Force

N

Torque . Nm

Record t r ace d e f l e c t i o n nun

S t r a i n us .

Stress x l O 5

N / m

2

98 29.4 10 91.7 73.4

196 58.8 20 183.5 146.8

215.6 64.7 21 192.7 154.2

235.2 70.6 22.5 206.4 165.1

254.8 76.4 24 220.2 176.2

274.4 82.3 26 238.5 190.8

Record No. - 2

A p p l i e d Force

N

Torque Nm.

Record t r ace d e f l e c t i o n mm

S t r a i n us.

Stress

x l O 5 N/m 2

19.6 5.9 7 64.2 51.4

39.2 11.8 15 137.6 110

58.8 17.7 . 22 201.8 161.5

68.6 20.6 26 238.5 190.8

78.4 23.5 29 266 212.8

88.2 26.5 32 293.6 234.9

107.8 32.3 38 348.6 278.9

117.6 35.3 42 385.3 308.2

132 Record No. - 3

A p p l i e d Force

N

Torque . Nm

Record t r ace d e f l e c t i o n mm

S t r a i n us.

Stress

x l O 5 N/m 2

98 29.4 9 275.2 220.2

117.6 35.3 10 305.8 244.6.

137.2 41.2 12 367 293.6

156.8 47 14 428 .1 342.5

166.6 50 .15 458.7 367

176.4 52.9 18 550.5 440.4

196 58.8 18.5 565.7 452.6

205.8 61.7 19 581 464.8

215.6 64.7 20 611.6 489.3

Record No. - 4

A p p l i e d Force

N •

Torque Nm.

Record t r a c e d e f l e c t i o n mm

S t r a i n us.

Stress

x l O 5 N/m 2 -

49 14.7 3.5 321.1 333.6

147 4 4 . 1 13.5 1238.5 1286.8

245 73.5 23 2110 2192.4

264.6 79.4 25 2293.6 2383

284.2 85.3 27 2477 2573.6

303.8 91 .1 29 2660.5 2764;3

133 Record No. - 5

A p p l i e d Force

N

Torque Nm

Record t r ace d e f l e c t i o n mm

S t r a i n us .

Stress

x l O 5 N/m 2

98 29.4 7.5 688 714.9

147 4 4 . 1 11.5 1055 1096.2

166.6 50 12.5 1146.8 1191.5

186.2 55.9 16 1467.9 1525.1

205.8 61.7 .18 1651.4 1715.8

Record No. - 6

A p p l i e d Force

N

Torque Nm.

Record t r a c e d e f l e c t i o n mm

S t r a i n us .

Stress

x l O 5 N/m 2

49 . 14.7 3.5 321.1 333.6

147 4 4 . 1 . 13 1192.6 1239.2

245 73.5 22.5 2064.2 2144.7

264.6 79.4 24.5 2247.7 2335.4

284.2 85.2 27 2477 2573.6

303.8 91 .1 29 2660.5 2764.3

134

Record No. - 7

App l i e d Force

N

Torque Nm

Record t r a c e d e f l e c t i o n mm

S t r a i n us.

S t r e s s x l O 5 N/m2

49 14.7 4 366.9 381.3

147 44.1 13 1192.7 1239. 2*.

245 73.5 22 2018.3 2097

264.6 79.4 24 2201.8 2287.7

284.2 85.2 •25.5 2339.4 2430.7

303.8 91.1 26.5 2431.2 2526

Record No. - 8

A p p l i e d Force

N

Torque Nm.

Record t r a c e d e f l e c t i o n mm

S t r a i n us. •

S t r e s s x l O 5 N/m2

98 29.4 3 917.4 953.2

147 44.1 3.5 1070.3 1112

166.6 50 4 1223.2 1271

186.2 55.9 4.5 1376.1 1429.8

205.8 61.7 . 5 1529 1588.7

225.4 67.6 5.5 1682 1747.5

245 73.5 6 1834.8 1906.4

135

Record No. - 9

Ap p l i e d Force

N

Torque . . Nm

Record t r a c e d e f l e c t i o n mm

S t r a i n us.

S t r e s s x l O 5 N/m2

19.6 5.9 2 61.1 63.5

39.2 11.7 4 122.3 127

58.8 17.6 8 244.6 254.2

78.4 23.5 12 367 381.3

98 29.4 17. 520 540.1

117.6 35.3 22 672.8 699

Record No. - 10

A p p l i e d Force

N

19.6

39.2

49

58.8

Torque Nm.

5.9

11.7

14.7

17.6

Record t r a c e d e f l e c t i o n mm

6.5

22

30.5

41

S t r a i n us.

59.6

201.8

279.8

376.1

S t r e s s x l O 5 N/m2

62

209.7

290.7

390.8

136

Record No. - 11

A p p l i e d Force

N

Torque Nm •

Record t r a c e d e f l e c t i o n mm

S t r a i n us.

S t r e s s x l O 5 N/m2

19.6 5.9 2.5 76.5 61.1

29.4 8.8 3.5 107 85.6

49 14.7 6 183.5 146.8

68.6 20.6 9 275.2 220.2

78.4 23.5 10 305.8 244.6

88.2 26.5 11 336.4 269.1

107.8 32.3 12.5 382.2 305.8

Record No. - 12

A p p l i e d Force

N

Torque Nm.

Record t r a c e d e f l e c t i o n nun

S t r a i n us.

S t r e s s x l O 5 N/m2

19.6 5.9 3 9.17 7.3

39.2 11.7 6.5 19.9 15.9

49 14.7 8.5 26 20.8

58.8 17.6 10 30.6 24.5

78.4 23.5 13.5 .41.3 33

J'

137 Record No. - 13

A p p l i e d Force

N

Torque Nm

Record t r a c e d e f l e c t i o n mm

S t r a i n us.

S t r e s s x l O 5 N/ m

2

49 14.7 8.5 259.9 270

68.6 20.6 14 428.1 444.8

88.2 26.5 19.5 596.3 619.6

107.8 32.3 25 764.5 794.3

127.4 38,2 •30 917.4 953.1

Record No. - 14

App l i e d F o r c e

N

98

117.6

137.2

156.8

Torque Nm.

29.4

35.3

41.1

47

Record t r a c e d e f l e c t i o n mm

46

55

64

S t r a i n us.

140.8

168.4

195.9

S t r e s s x l O 5 N/m2

112.7

72.5 222

134.7

156.7

177.6

138 Record No. - 15

A p p l i e d Force

N

Torque Nm

Record t r a c e d e f l e c t i o n mm

S t r a i n us.

S t r e s s x l O 5 N/m2

19.6 5.9 9.5 29 23.2

39.2 11.7 20 61.1 •

48i 9

58.8 17.6 29.5 90.2 72.1

78.4 23.5 40 122.3 97.9

88.2 26.5 •45.5 139.1 111.3

98 29.4 51 155.9 124.8

117.6 35.3 64 195.7 156.6

Record No. - 16

A p p l i e d Force

N

Torque Nm.

Record t r a c e d e f l e c t i o n mm

S t r a i n us.

S t r e s s x l O 5 N/m2

98. 29.4 24 73.4 58.7

117.6 35.3 31 •94-; 8 75.8

137.2 41.2 38 116.2 93

156.8 47 46 140.6 112.5

•

139 Record No. - 17

A p p l i e d Force

N

Torque Nm

Record t r a c e d e f l e c t i o n mm

S t r a i n us.

S t r e s s x l O 5 N/m2

19.6 5.9 2 6.1 4.9

39.2 11.7 6 18.3 14.7

58.8 17.6 13.5 41.3 33

78.4 23.5 21.5 65.8 52.6

98 29.4 30 91.7 73.4

117.6 35.3 40 122.3 97.9

Record No. - 18

A p p l i e d Force

N

Torque Nm.

Record t r a c e d e f l e c t i o n mm

S t r a i n us.

S t r e s s x l O 5 N/m2

19.6 5.9 2.5 22.9 23.8

39.2 11.7 10.5 96.3 100

58.8 17.6 22 201.8 209.7

68.6 20.6 22 201.8 209.7

78.4 23.5 24.5 224.8 233.5

98 29.4 31.5 289 300.3

Record No. - 19 140

A p p l i e d Force

N

* Torque . Nm

Record t r a c e d e f l e c t i o n mm

S t r a i n us.

S t r e s s x l O 5 N/m2

98 29.4 3 275.2 286

147 44.1 5.5 504.6 524.3

166.6 50 6 550.5 571.9

186.2 55.8 7 642.2 667.2

205.8 61.7 8 733.9 762.6

225.4 67.6 9 825.7 857.9

245 73.5 9.5 871.5 905.5

. r

Record No. - 20

A p p l i e d Force

N

Torque Nm.

Record t r a c e d e f l e c t i o n mm

S t r a i n us.

S t r e s s x l O 5 N/m2

49 14.7 2 183.5 190.6

147 44.1 6.5 596.3 619.6

. 245 73.5 10.5 963.3 1000.9

264.6 79.4 - - —

-

•

i.

141

Record No. - 21

App l i e d Force

N

Torque Nm

Record t r a c e d e f l e c t i o n mm

S t r a i n us.

S t r e s s x l O 5 N/m2

98 29.4 11 336.4 349.5

147 44.1 18 550.5 571.9

166.6 50 20 611.6 635.5.

186.2 55.9 23 703.3 730.8

205.8 61.7 26 795.1 826.1

225.4 62.6 29 886.8 921.4

235.2 70.5 31 948 985

245 73.5 34 . 1039.7 1080.3

.'• 1 T !

Record No. - 22

A p p l i e d Force

N

Torque Nm.

Record t r a c e d e f l e c t i o n mm

S t r a i n us.

S t r e s s x l O 5 N/m2

49 14.7 4.5 137.6 143

68.6 20.6 7 214 222.4

88.2 26.5 10 305.8 317.7

107.8 32.3 13 397.5 413

205.8 61.7 21.5 657.5 683.1

142

Record No. - 23

Ap p l i e d Force

N

Torque Nm

Record t r a c e d e f l e c t i o n mm

S t r a i n us.

S t r e s s x l O 5 N/m2

98 29.4 10.5 96.3 77

117.6 35.3 19 174.3 139.4

137.2 41.2 21.5 197.2 157.8

156.8 47 24.5 224.8 179.8

176.4 52.9 28 256.9 205.5

186.2 55.9 29 266 212.8

196 58.8 31 284.4 227.5

205.8 61.7 34 311.9 249.5

Record No. - 24

Ap p l i e d Force

N

Torque Nm.

Record t r a c e d e f l e c t i o n mm

S t r a i n us.

S t r e s s x l O 5 N/m2

19.6 5.9 6 5.5 4.4

39.2 11.7 25 22.9 18.3

58.8 17.6 50 45.9 36.7

78.4 23.5 77 70.6 56.5

98 29.4 115 105.5 84.4

Record No. - 25 143

A p p l i e d Force

N

Torque . Nm

Record t r a c e d e f l e c t i o n mm

S t r a i n us.

S t r e s s x l O 5 N/m2

98 29.4 8 733.9 762

196 58.8 17.5 -1605.5 1668.1

215.6 64.7 20 1834.9 1906.4

264.6 79.4 24.5 2247.7 233514

Record No. - 26

A p p l i e d Force

N

Torque Nm.

Record t r a c e d e f l e c t i o n mm

S t r a i n us.

S t r e s s x l O 5 N/m2

98 29.4 10.5 321.1 256.9

117.6 35.3 12 367 293.6

137.2 41.1 14 428.1 342.5

147 44.1 15 458.7 367

156.8 47 16 489.3 391.4

176.4 52.9 16 489.3 391.4

186.2 55.9 17.5 535.1 428.1

144

Record No. - 27 •

A p p l i e d Force

N

Torque . Nm

Record t r a c e d e f l e c t i o n mm

S t r a i n us.

S t r e s s x l O 5

N/m2

98 29.4 9.5 290.5 232.4

117.6 35.3 11 336.4 269.1

127.4 38.2 11.5 351.7 281.3

137.2 41.1 12 367 293.6

156.8 47 .13.5 412.8 330.3

Record No. - 28

A p p l i e d Force

N

Torque Nm,

Record t r a c e d e f l e c t i o n mm

S t r a i n us.

S t r e s s x l O 5 N/m2

98 29.4 17.5 160.5 128.4

196 58.8 34 311.9 249.5

215.6 64.7 39 357.8 286.2

-

145

Chapter 5 - D i s c u s s i o n of R e s u l t s

5.1 Determination of A x i a l working s t r e s s f o r l i n k 5 - A t h e o r e t i c a l approach

Load and s t r e s s v a r i a t i o n i n l i n k 5 have been t h e o r e t i c a l l y determined

i n chapter 3. Before comparing the experimental and t h e o r e t i c a l s t r e s s

v a r i a t i o n i n d e t a i l , the f o l l o w i n g a n a l y t i c a l a n a l y s i s i s performed on the

l i n k to determine the s a f e working a x i a l s t r e s s l i n e assuming t h a t the machine

i s designed f o r a crank speed of 4000 r.p.m. and the l i n k s a r e p e r f e c t l y

e l a s t i c , homogeneous and i s o t r o p i c . The s u r f a c e roughness and e c c e n t r i c i t y

i n the bore axes have been neglected. The j o i n t s a r e taken to be i d e a l

without any p l a y ( p l a y i n the j o i n t s i n t r o d u c e s a mechanical e r r o r of an

a p p r e c i a b l e amount - because of hydrodynamic a c t i o n of the l u b r i c a n t the

p i n a x i s w i l l not touch the end bore c i r c l e when the mechanism i s i n motion).

From F i g . (5.1)-b, the minimum c r o s s s e c t i o n a l area f o r l i n k 5 i s approximately -5 2

4 x 10 m ( c r o s s - s e c t i o n XX) w h i l e the maximum c r o s s s e c t i o n a l area i s -5 2

approximately 6.32 x 10 m ( c r o s s - s e c t i o n Z Z ) . The r a t i o of maximum c r o s s -

s e c t i o n a l area to minimum c r o s s - s e c t i o n a l area being 1.58. The a c t u a l

a x i a l l a o d i n g diagram f o r the l i n k i s shown i n F i g . (5.2a) f o r a crank speed

of 4000 r.p.m. The s t r e s s v a r i a t i o n f o r c r o s s - s e c t i o n XX a t the same crank

speed i s shown i n F i g . ( 5 . 2 b ) . The s t r e s s waveform i s complex and an i d e a l ­

i z e d model f o r the s t r e s s v a r i a t i o n i s n e c e s s a r y to determine the f a t i g u e

s t r e n g t h and the s a f e working s t r e s s l i n e . Assuming a s i n u s o i d a l model, the

s t r e s s v a r i a t i o n can be expressed by the f o l l o w i n g equation:

2 n t (5.1) a = o + o" s i n m r T

where a mean s t r e s s m

ex. v a r i a b l e s t r e s s r T time f o r one complete c y c l e = 0.015 s (4000 r.p.m.)

s t r e s s v a r i a t i o n

; 146 F i g 5.1

l i n k 3 and Link 5 (clearances are extremely exaggerated)

i U

14 U

ATOSS section of the 7 BOLT

A Cross section UU

'A A >

'A 23 Cross section XX

A Scale x2

147

A

u

o v .

a. f— $4 « 5

O L D

o LO £ a.

H3 D J a

0)

CM

o a CO L i )

o O O if)

If) v?

CO o -4—> Q.

If)

148

S u b s t i t u t i n g the numerical v a l u e s eq. (5.1) y i e l d s :

a = -75 x 10° + 478 x 10 s i n 2 n t

0T015 (4000 r.p.m.)

which i s shown i n F i g . (5.3)

S t r e s s r a t i o i s determined as:

n crm,„ 553 x 10 . _„ R = mm = - = -1.37 Jmax 103 x 10

The f a c t o r of s a f e t y F.S can now be c a l c u l a t e d as:

max y-p F.S

F.S. = 17.6 x 1 0 7 = 3.18 553 x 1 0 5

where the y i e l d point s t r e n g t h o* = 17.6 x 10 i s taken from F i g . (4.8)

There a r e a number of d i f f e r e n t e m p i r i c a l f a i l u r e equations d e f i n i n g

the r e l a t i o n between the v a r i a b l e and mean s t r e s s e s . Three of the most

commonly used r e l a t i o n s a r e employed to c a l c u l a t e the endurance s t r e n g t h <j

as i n the f o l l o w i n g :

a - Gerber p a r a b o l i c r e l a t i o n ;

c re/ + VW = 1

7 2 where CT = u l t i m a t e t e n s i l e s t r e n g t h = 22.4 x 10 N/m (From F i g . ( 4 . 8 ) )

u l t

CT = + 4.78 x 1 0 7 N/m2

e —

149

b - Modified Goodman R e l a t i o n ;

m + u l t

+ 4.62 x 10 N/m

Soderberg r e l a t i o n

m yp

CT = + 4.58 x 1 0 7 N/m2

e

The s a f e working a x i a l s t r e s s l i n e , employing Soderberg's c r i t e r i a i s

shown i n F i g . ( 5 . 3 ) . The a x i a l s t r e s s a t a crank speed of 4000'r.p.m. can

h a r d l y be approximated by Gerber's c r i t e r i a and i s above the approximate

l i n e s of f a i l u r e by Soderberg and modified Goodman methods, w h i l e the

a x i a l s t r e s s a t a crank speed of 25000'^.p.m. i s s l i g h t l y under the conser- /<:

v a t i v e s a f e s t r e s s l i n e approximated by Soderberg c r i t e r i a . A good d i s c u s s i o n

of the above procedure can be found i n ( 20 ) ( 26 ) and ( 5 ) . I t

should be noted t h a t the v a l u e s c a l c u l a t e d a r e only rough e s t i m a t e s . No

standard b a s i s f o r o b t a i n i n g working f a t i g u e s t r e s s r e l a t i o n s has been

u n i v e r s a l l y accepted. The f a t i g u e s t r e n g t h i s a l s o a f f e c t e d by the c l e a r a n c e

between the p i n and hole, the d i s t r i b u t i o n of shear s t r e s s around the hole

boundary, l u b r i c a t i o n , p i n m a t e r i a l and p i n bending e f f e c t s , s u r f a c e c o n d i t i o n s ,

p r i o r overloads, environmental e f f e c t s , m a t e r i a l and manufacturing e f f e c t s .

The c a l c u l a t e d endurance l i m i t s t r e s s , o" , i s the maximum completely r e v e r s i n g

s t r e s s t h a t the l i n k can s u s t a i n f o r an u n l i m i t e d number of c y c l e s without

f a t i g u e f a i l u r e . I f the completely r e v e r s e d s t r e s s i s h i g h e r than a , the

i I O U

1 a:

4 +» r-l

•r. HH / CM'

a.

10

/ 10 £1

/ 11

\ I.)

VJ O- 3JW o

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t 3

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ri

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151

f a i l u r e can be expected to take p l a c e a f t e r some f i n i t e number of c y c l e s .

The higher the s t r e s s the fewer w i l l be the c y c l e s before f a i l u r e can be

expected.

5.2 T h e o r e t i c a l and experimental a x i a l s t r e s s range v a r i a t i o n w i t h the

crank speed i n l i n k 5

The v a r i a t i o n of the experimental a x i a l s t r e s s range f o r l i n k 5 w i t h

the crank speed and w i t h the square of the crank speed a r e presented i n

F i g u r e s 4.9 and 4.13 r e s p e c t i v e l y . Due to gauge i n s t a l l a t i o n and s t r u c t ­

u r a l f a i l u r e s (due to s l i p ) , c a u s i n g s e v e r e mechanical damage; the maximum

crank arm speed could not be i n c r e a s e d beyond about 25000 r.p.m. although

v a r i o u s u n s e c c e s f u l attempts have been made to reach 3000 r.p.m. or more.

The speed f l u c t u a t i o n e f f e c t has been n e g l e c t e d ( a s e x p l a i n e d i n chapter 4,

+ 0.08 experimental speed f l u c t u a t i o n c o e f f i c i e n t has been regarded as

being of no importance). The b a s i c f a c t o r s i n t r o d u c i n g e r r o r a r e i d e n t i f i e d

as f o l l o w s ;

1 - S e n s i t i v i t y and a c c u r a c y of the measuring d e v i c e s employed i n the

experimental set-up.

2 - E f f e c t of the h o s t i l e enivronment

3 - E f f e c t of the a d d i t i o n a l moment of i n e r t i a introduced by the

s t r a i n - g a u g e s and t h e i r l e a d s on the l i n k s

4 - E r r o r i n experimental determination of the c r o s s - s e c t i o n a l a r e a s

and v a l u e s of Young's moduli.

5 - P e r s o n a l e r r o r i n i n t e r p r e t i n g the data (u.v. T r a c e s ) .

I n g e neral the experimental r e s u l t s a r e i n good agreement w i t h the

c a l c u l a t e d a x i a l s t r e s s range v a l u e s f o r l i n k 5. By a p p l y i n g method of

l e a s t squares the experimental a x i a l s t r e s s range curve i s approximated by

the f o l l o w i n g s t r a i g h t l i n e

100 n

y = 38.6 x - 7.7

152

5 2 where y = experimental a x i a l - s t r e s s range -10 N/m

4 2

x = square o f the crank speed -10 ( r a d / s )

The t h e o r e t i c a l v a r i a t i o n i s i n the form;

y = 34.6x a d e t a i l e d examination of F i g . 4.9 i s presented i n the f o l l o w i n g t a b l e :

T able 5.1

A B approximate approximate T h e o r e t i c a l approximate Experimental . crank speed a x i a l s t r e s s range a x i a l s t r e s s range |B-A|xlOO

r.p.m. x 1 0 5 N/m2 x 1 0 5 N/m2

1100 45 45 0

1300 63 55 12.7

1410 73 69 5.5

1600 95 104 9.5

1700 108 106 1.9

1800 120 128 6.7

1970 143 143 0

2030 154 171 11

2110 164 194 18.3

2310 197 216 9.6

2410 216 242 12

Average v a l u e of | xlOO = 7.9

The maximum d e v i a t i o n occurs a t a crank speed o f 2110 r.p.m. (18.3%, 5 . 2

corresponding to a d i f f e r e n c e of 30 x 10 N/m between t h e o r e t i c a l and

experimental s t r e s s range). The experimental v a l u e s c o n s i s t e n t l y tend to

be higher than c a l c u l a t e d v a l u e s a f t e r 1970 r.p.m. A comparison of theor­

e t i c a l and experimental v a r i a t i o n of s t r e s s i n a complete c y c l e (1 rev = 360°)

f o r a crank speed of 1534 r.p.m. i s shown i n F i g . 4.16. The c l o s e resemblance

153

between the waveforms i s s i g n i f i c a n t . An approximate s h i f t of 20 i s

observed a t the two peak t e n s i l e s t r e s s v a l u e s , between the two waveforms.

The sample of the recorded a x i a l s t r e s s waveform over one c y c l e shown i n

F i g . 4 . I S i s i n b e t t e r agreement w i t h the c a l c u l a t e d waveform than the

experimental waveform presented and compared w i t h the c a l c u l a t e d waveform

by Alexander and Lawrence( 15), f o r bending s t r a i n v a r i a t i o n i n the coupler

o f a. model four-bar p l a n a r mechanism.

' 154

5.3 C a l c u l a t i o n of e c c e n t r i c i t y from experimental data, f o r l i n k s 5 and 3

a - L i n k 5

From F i g . 4.4 - s t a t i c bending s t r a i n t e s t f o r l i n k 5, the equation

of the l i n e approximating a x i a l s t r e s s v a r i a t i o n w i t h the a p p l i e d torque i s

a ax = 12.7

. „ . , _ _ A x i a l f o r c e where cr = a x x a l s t r e s s on the c r o s s sectxon = ax c r o s s - s e c t i o n a l area

T = Torque a p p l i e d to the crank arm v i a the l e v e r

and the equation of the s t r a i g h t l i n e approximating the bending s t r e s s

v a r i a t i o n w i t h the a p p l i e d torque i s :

a J T £ = 26.6

M where cr = bending s t r e s s = C

I

where I = Moment of i n e r t i a of the c r o s s s e c t i o n

c = Maximum d i s t a n c e from the n e u t r a l a x i a M = F .d ax

where d = e c c e n t r i c i t y

The r a t i o of the bending s t r e s s to a x i a l s t r e s s i s

K = °"b = 2 CTax

The e c c e n t r i c i t y d can now be c a l c u l a t e d a s :

K = CTb = 2 = d.c.A. (5 2 ) CTax 1

155

where A = c r o s s s e c t i o n a l area

eq. (5.2 ) can be w r i t t e n as

d = KI (5.3) cA

assuming the c r o s s s e c t i o n a l area to be a p e r f e c t rectangle.;-

I = b h 3

12

where b = base le n g t h of the r e c t a n g l e

h = h e i g h t of the r e c t a n g l e

c = h/2

eq. (5.3) becomes

d = Kh (5.4) 6

s u b s t i t u t i n g the a p p r o p r i a t e v a l u e s :

d = H ~ 2.8 mm 3

b - L i n k 3

S i m i l a r l y from F i g 4.R s t a t i c bending s t r a i n t e s t f o r l i n k 3

a a ax = 4 and _b = 8.75 T T

k = o"b = 2.2

a ax

d = Kh = 0.36h = 4.3 mm 6

156

I n the above c a l c u l a t i o n s i t i s assumed t h a t the m a t e r i a l f o l l o w s Hooke's

law and the magnitude of the s t r e s s i s p r o p o r t i o n a l to the d i s t a n c e from

the n e u t r a l a x i s .

5.4 E c c e n t r i c i t y - dynamic c a s e

The v a r i a t i o n of a x i a l and bending s t r e s s e s i n l i n k s 5 and 3 w i t h

the crank speed were shown i n F i g u r e s 4.!9 , 4.11 , 4..12 and 4.10 r e s p e c t i v e l y .

The r a t i o K ^ o f the experimental bending s t r e s s ranges to experimental a x i a l

s t r a i n ranges f o r both l i n k s a r e given i n the f o l l o w i n g t a b l e , Table 5.2

a and b.

Tab l e 5.2-a

crank speed K dynamic r.p.m. l i n k 5

1080 1.9

1280 2

1420 2.2

1610 2

1800 2.5

2050 2.4

2400 2.4

K dynamic average

= 2.2

Tab l e 5.2-b

crank speed k dynamic r.p.m. l i n k 3

1082 7.3

1320 3.3

1440 3.5

1560 3.4 K dynamic average = 3.9

1800 3.4

2040 2.6

157

Average value of K dynamic f o r l i n k 5 i s i n good agreement w i t h the K

v a l u e f o r the s t a t i c c a s e w h i l e the average value of K dynamic f o r l i n k 3

i s about 1.77 times of the K v a l u e obtained f o r the s t a t i c c a s e

5. 5 Determination of A x i a l working s t r e s s f o r l i n k 3 - A t h e o r e t i c a l approach

Applying the same assumptions and method shown i n s e c t i o n 5.1., the

working s t r e s s i s determined f o r the minimum c r o s s s e c t i o n a l area ( c r o s s

s e c t i o n UU, F i g . 5.1) as i n the f o l l o w i n g ;

The loading diagram and a x i a l c y c l i c s t r e s s v a r i a t i o n a r e shown i n F i g . 5.4

f o r the c r o s s s e c t i o n a t a crank speed of 4000 r.p.m. -5 2

A = c r o s s s e c t i o n a l area 11 x 10 m

CT = 302 x 1 0 5 N/m2

max a . = -88 x 1 0 5 N/m2

min a 107 x 1 0 5 N/m2

m °~r = 195 x 1 0 5 N/m2

2a = 390 x 1 0 5 N/m2

'r 0.015 s

u s i n g eq. (5.1) the a c t u a l s t r e s s p a t t e r n i s approximated by the f o l l o w i n g

s i n u s o i d a l model:

CT = 107 x 1 0 5 + 195 x 1 0 5 s i n 2 n t 0.015

which i s shewn i n F i g . (5.4)

The s t r e s s r a t i o i s :

min R = = -0.3

CT max

158

From F i g (4.8)

o- = 11.4 x 1 0 7 N/m2

CTult = 1 2 , 4 x 1 q 7 N / i " 2

The f a c t o r of s a f e t y can be c a l c u l a t e d as

max F.S

F.S * 3.77

7 2 Soderberg r e l a t i o n g i v e s an endurance s t r e n g t h of 2.15 x 10 N/m . c&

7 2

c a l c u l a t e d by modified Goodman r e l a t i o n i s 2.13 x 10 N/m . The working

s t r e s s diagram i s shown i n F i g . (5.5 ) . A x i a l s t r e s s a c t i n g on the

c r o s s s e c t i o n i s above the Soderberg l i n e . C r o s s s e c t i o n MM . ( F i g .

5.1) i s i l s o represented on the same diagram.

5.6 T h e o r e t i c a l and experimental a x i a l s t r e s s range v a r i a t i o n w i t h the

crank speed i n l i n k 3.

The v a r i a t i o n of the experimental a x i a l s t r e s s range f o r l i n k 3 w i t h

the crank speed and w i t h the square of the crank speed a r e presented i n

F i g u r e s 4.12 and 4.13: r e s p e c t i v e l y . Crank speed could not be i n c r e a s e d

beyond about 2100 r.p.m. (due t o gauge i n s t a l l a t i o n f a i l u r e s , and d r i f t i n

the u.v. t r a c e s ) . By ap p l y i n g the method of l e a s t squares the experimental

a x i a l s t r e s s range curve i s approximated by the s t r a i g h t l i n e

y = 21.8x + 11.4

5 , 2 where y = experimental a x i a l s t r e s s range -10 N/m

4 2 x = square of the crank speed -10 ( r a d / s )

159

the ' t h e o r e t i c a l v a r i a t i o n i s i n the form:

y = 21.4x

a d e t a i l e d examination of F i g . 4.12 i s presented i n the f o l l o w i n g t a b l e :

T able 5.3

approximate crank speed

r.p.m.

1100

1260

1430

1550

1790

1910

2080

2130

Approximate t h e o r e t i c a l a x i a l s t r e s s range

5 . 2 10 N/m

27

36

47

57

75

85

100

105

B Approximate experimental

a x i a l s t r e s s range

1 0 5 N/m2

24

57

70

74

83

94

114

123

B-A 100

11.1

58.3

49

30

10.6

10.6

14

17.1

average value of B-A| xlOO = 25 I A I

which i s about 3 times higher than t h a t of l i n k 5. The experiment v a l u e s

a r e g e n e r a l l y higher than the c a l c u l a t e d v a l u e s . The maximum d e v i a t i o n occurs

a t a crank speed of 1260 r.p.m. ( 5 8 . 3 % ) . A comparison of t h e o r e t i c a l and

experimental c y c l i c a x i a l s t r e s s v a r i a t i o n i s shown i n F i g . 4.15

5.7 Bending s t r e s s e s i n a d i r e c t i o n normal to plane of the mechanism

The v a r i a t i o n of peak to peak bending s t r e s s e s w i t h the crank speed

f o r l i n k s 3 and 5 are shown i n F i g u r e s 4.30 and 4.1-1 r e s p e c t i v e l y .

160

The bending s t r e s s range curves can be approximated by the s t r a i g h t l i n e s :

y = 37x + 150 ( l i n k 3)

y = 104x - 58 ( l i n k 5)

where aga i n :

5 2 y = bending a x i a l s t r e s s range - 10 N/m

2 x = square of crank speed -10 ( r a d / s )

At a crank speed of 4000 r.p.m. the expected bending s t r e s s range v a l u e s 5 2 5 2 f o r the l i n k s 3 and 5 can be c a l c u l a t e d as 800 x 10 N/m and 1767 x 10 N/m

r e s p e c t i v e l y . The expected a x i a l s t r e s s ranges a t the same crank speed f o r the

l i n k s a r e 395 x 1 0 5 N/m2 ( l i n k 3) and 670 x 1 0 5 N/m2 ( l i n k 5 ) ( v a l u e s based

on experimental d a t a ) . The v a r i a b l e s t r e s s v a l u e s a r e w e l l above the endur­

ance s t r e n g t h of the l i n k s . The combined e f f e c t of bending and a x i a l

s t r e s s e s (method of s u p e r p o s i t i o n ) a t a crank speed of 4000 r.p.m. would

i n d i c a t e a s t a t e of f a i l u r e .

5.8 S t r e s s d i s t r i b u t i o n i n the l i n k s

The main o b j e c t i v e of the t e s t e x p lained i n s e c t i o n 4.4^:3 was to determine

whether the bending s t r e s s e s a r e due to an i n i t i a l permanent l i n k deformation,

an o f f s e t , or due to the p i n s , t o l e r a n c e and/or c l e a r a n c e e f f e c t s or" other

misalignments which a r e f a c t o r s e x t e r n a l to the l i n k s themselves. A d e t a i l e d

examination of F i g u r e s 4.6 and 4.7 a r e presented i n T a b l e s (5.4 - 7 ) .

R e s u l t s of the t e s t s f o r both l i n k s show t h a t the bending s t r e s s ( z - d i r e c t i o n )

a r e due to e f f e c t s which a r e e x t e r n a l to the l i n k s . The b a s i c f a c t o r s i n t r o ­

ducing e r r o r to the r e s u l t s a r e i d e n t i f i e d as f o l l o w s :

1 - E f f e c t s of d e v i a t i o n i n the crank angle (90° + 5°)

2 - F r i c t i o n i n the bearings

3 - E f f e c t s of e l a s t i c i t y of the s h a f t s and l i n k s

The p o s s i b l e bending mechanisms a r e shown i n F i g u r e s (5.6) and (5.7)

161

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164

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167 I

| F 1 (b.excess c learance

I

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7

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ink shape

F i g . 5 . r

M

71 F i g . 5.7

c .s ingle or double. pin deformation

168

5.9 General C o n s i d e r a t i o n s

During the a c t u a l weaving operation the s t r e s s e s would i n c r e a s e due to

the e x t e r n a l r e s i s t a n c e introduced by the threads on the combs and needle

arms. The va l u e of t h i s expected i n c r e a s e i s unknown. The hardness and

t e n s i l e t e s t s performed on the connecting rods showed th a t t h e r e i s a

d i f f e r e n c e between m a t e r i a l p r o p e r t i e s of the two l i n k s although they were

expected to be the same (phosphor bronze a l l o y ) . Mechanical damage ( s c r a t c h e s

i n s i d e the l i n k bores and fl a n g e s i d e s ) i s detected on both l i n k s . The

e f f e c t of f r i c t i o n has completely been ne g l e c t e d i n the a n a l y t i c a l c a l c u l ­

a t i o n s . S t r e s s waveform p a t t e r n i s complex i n s t r u c t u r e f o r both l i n k s and

i s very d e s t r u c t i v e i n nature from a - f a t i g u e point of view.

169

Chapter 6 - Conclusion

Bending s t r e s s e s i n a d i r e c t i o n normal to the plane of the mechanism

a r e present i n both of the l i n k s i n v e s t i g a t e d and have a dominating e f f e c t .

The a x i a l c y c l i c and peak to peak s t r e s s v a r i a t i o n i n both l i n k s i s g e n e r a l l y

i n good agreement w i t h the c a l c u l a t e d v a l u e s . The bending s t r e s s e s normal

to the plane of the mechanism a r e due to misalignment e f f e c t s which a r e

e x t e r n a l to the connecting rods themselves. The dominating c h a r a c t e r i s t i c

of the dynamic bending s t r e s s e s has been confirmed w i t h s t a t i c t e s t s

c a r r i e d out on the mechanism. Even by n e g l e c t i n g the bending s t r e s s e f f e c t s

on the l i n k s the present design i s found to be u n s a t i s f a c t o r y a t a crank

speed o f 4000 r.p.m. s i n c e the a x i a l s t r e s s v a l u e s a r e above the working

a x i a l s t r e s s l i n e approximated by the Soderberg c r i t e r i a . I n the presence

of bending s t r e s s e s f a t i g u e f a i l u r e can be expected to occur a t approximately

2000 r.p.m. However i f bending s t r e s s e s a r e n e g l i g i b l e the present design

can be used up to a crank speed of 2500 r.p.m. (See Appendix A3)

To i n c r e a s e the speed of the mechanism the f o l l o w i n g general b a s i c

major p o i n t s a r e suggested:

1. A d e t a i l e d d e f l e c t i o n t o l e r a n c e and c l e a r a n c e a n a l y s i s to reduce

the bending s t r e s s e s due to misalignments t h e o r e t i c a l l y to zero

which would a u t o m a t i c a l l y i n c r e a s e the crank speed range up to

2500 r.p.m.

2. F a t i g u e t e s t s on the l i n k s to determine the experimental endurance

s t r e n g t h .

3. The s t r e s s c o n s t r a i n t s a r e determined. Any i n c r e a s e i n the c r i t i c a l

c r o s s s e c t i o n s of the connecting rods w i l l decrease the s t r e s s e s , and

t h e r e f o r e would y i e l d a higher o p e r a t i n g speed.

4. I t was found t h a t extreme, a x i a l s t r e s s e s i n a complete c y l e of

input crank r o a t a t i o n o ccurs a t d i f f e r e n t a n g les of input crank

170

r o t a t i o n f o r the two connecting rods. Any s l i g h t changes i n

l i n k l e n g t h s and geometry to decrease the extreme angular

a c c e l e r a t i o n v a l u e s f o r l i n k s 4 and 6 w i l l a c c o r d i n g l y decrease

the a x i a l s t r e s s e s on the two connecting rods and change the

c y c l i c a x i a l s t r e s s p a t t e r n .

Without changing the c o n f i g u r a t i o n of the mechanism, any

s u i t a b l e m a t e r i a l change w i t h a higher endurance and y i e l d point

l i m i t w i l l a c c o r d i n g l y permit an i n c r e a s e i n the crank speed.

171

Appendix A - l

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172

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REFERENCES

( 1 ) Chen, F.Y. An a n a l y t i c a l method for synthesizing the four-bar crank-rocker mechanism. Journal of Engineering for Industry, Trans. ASME February 1969. p. 45-54

(2 ) Lord, P.R. and Mohamed, M.H. An analysis of the picking mechanism of a t e x t i l e loom. Journal of Engineering for Industry, Trans. ASME. May 1975. p. 385-390

( 3 ) Bevan,.'T. The Theory of Machines. Longmans, Green & Co. Ltd., 1964

( 4 ) Conte, F.L., George, G.R., Mayne, R.W., and Sadler, J.P. Optimum Mechanism Design combining Kinematic and Dynamic-Force considerations. Journal of Engineering for Industry, Trans. ASME. May 1975, p. 662-670

( 5 X Spotts, M.F. Design of Machine Elements (4th E d i t i o n ) . Brentice-Hall Inc. Englewood C l i f f s , N.J., U.S.A. 1971

( 6 ) Dove, R.C. and Adams, P.H. Experimental Stress Analysis and Motion Measurement. Charles E. M e r i l l Books. Inc., Ohio, U.S.A. 1964

( 7 ) Marin, J . Mechanical Behaviour of Engineering Materials. Prentice-Hall Inc., Englewood C l i f f s , N.J., U.S.A. 1962

( 8 ) Green, W.G. Theory of Machines (2nd Edition) Blackie & Son Ltd., 1962

( 9^ Shigley, J.E. Theory of Machines (International Student Ed.) McGraw-H i l l Book Co. Inc. 1961

0-10) Tepper, F.R. and Lowen, G.G. Shaking Force Optimisation of Four-bar Linkage with Adjustable Constraints on Ground Bearing Forces. Journal of Engineering for Industry, Trans. ASME. May 1975, p 643-651

(11.) Wojcik, C.K. Synthesis of Four-bar Linkage Function generators by means of Equations of Motion. Journal of Engineering for Industry. Trans. ASME. May 1965. p. 170-176

(12 ) Maxwell, R.L. Kinematics and Dynamics of Machinery. Prentice H a l l , Inc. Englewood C l i f f s , N.J., U.S.A. 1960

(13 ) Prentis, J.M. Dynamics of Mechanical Systems. Longman Group Ltd., Great B r i t a i n 1970

188

(14) Morrison, J.L.M. and Crossland, B. An introduction to the Mechanics of Machines. Longman Group Ltd., Great B r i t a i n . 1970

(15) Alexander, R.M. and Lawrence, K.L. Experimentally Determined Dynamic Strains i n an E l a s t i c Mechanism. Journal of Engineering for Industry, Trans. ASME 1974. Paper No. 74 - DET - 33

(16) Dubowsky, S., and Young, S.C. An Experimental and An a l y t i c a l Study of Connection Forces i n High-Speed Mechanisms. Journal of Engineering for Industry, Trans. ASME 1974. Paper No. 74 - DET - 72

(17) Singer, F.L. Strength of Materials (Second E d i t i o n ) . Harper and Row, Publishers. 1962

(18) Roark, R.J. Formulas for Stress and Stra i n . (Fourth Edition) McGraw-H i l l Book Company 1965

(19) Holister, G.S. Experimental Stress Analysis. Cambridge University Press. 1967.

(20) Heywood, R.B. Designing Against Fatigue. Chapman and Hall Ltd., 1962

(21) Holowenko, A.R. Dynamics of Machinery. John Wiley & Sons. Inc. 1963

(22) Hinkle, R.T. Kinematics of Machines. Prentice-Hall, Inc. Englewood C l i f f s , N.J., U.S.A. 1953

(23) Rayevskii, N.P. The Measurement of Mechanical Parameters in Machines. Pergamon Press. Ltd., 1965

(24) Greenwood*, J.M. Stresses i n the-Reed and Weft Needle Arm. of a Weaving Loom. F i n a l Year Honours Project. May 1974. University of Durham. Department of Engineering Science.

(25) Imam, I . , and Sandor, G.N. High-Speed Mechanicsm Design - A General Ana l y t i c a l Approach. Journal of Engineering for Industry. Trans. ASME. May 1975. Paper No. 74 - DET - 50

(26) Fa i r e s , V.M. Design of Machine Elements (4th E d i t i o n ) . Macmillan Co. 1962, U.S.A.

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